MATH 532: Linear Algebra Chapter 5: Norms, Inner Products and Orthogonality
Greg Fasshauer
Department of Applied Mathematics Illinois Institute of Technology
Spring 2015
[email protected] MATH 532 1 Outline
1 Vector Norms
2 Matrix Norms
3 Inner Product Spaces
4 Orthogonal Vectors
5 Gram–Schmidt Orthogonalization & QR Factorization
6 Unitary and Orthogonal Matrices
7 Orthogonal Reduction
8 Complementary Subspaces
9 Orthogonal Decomposition
10 Singular Value Decomposition
11 Orthogonal Projections
[email protected] MATH 532 2 Vector Norms Outline
1 Vector Norms
2 Matrix Norms
3 Inner Product Spaces
4 Orthogonal Vectors
5 Gram–Schmidt Orthogonalization & QR Factorization
6 Unitary and Orthogonal Matrices
7 Orthogonal Reduction
8 Complementary Subspaces
9 Orthogonal Decomposition
10 Singular Value Decomposition
11 Orthogonal Projections
[email protected] MATH 532 3 Vector Norms Vector Norms
Definition Let x, y ∈ Rn (Cn). Then
n T X x y = xi yi ∈ R i=1 n ∗ X x y = x¯i yi ∈ C i=1
is called the standard inner product for Rn (Cn).
[email protected] MATH 532 4 Vector Norms
Definition
Let V be a vector space. A function k · k : V → R≥0 is called a norm provided for any x, y ∈ V and α ∈ R
1 kxk ≥ 0 and kxk = 0 if and only if x = 0,
2 kαxk = |α| kxk,
3 kx + yk ≤ kxk + kyk.
Remark The inequality in (3) is known as the triangle inequality.
[email protected] MATH 532 5 We will show that the standard inner product induces the Euclidean norm (cf. length of a vector).
Remark Inner products let us define angles via
xT y cos θ = . kxkkyk
In particular, x, y are orthogonal if and only if xT y = 0.
Vector Norms
Remark Any inner product h·, ·i induces a norm via (more later) p kxk = hx, xi.
[email protected] MATH 532 6 Remark Inner products let us define angles via
xT y cos θ = . kxkkyk
In particular, x, y are orthogonal if and only if xT y = 0.
Vector Norms
Remark Any inner product h·, ·i induces a norm via (more later) p kxk = hx, xi.
We will show that the standard inner product induces the Euclidean norm (cf. length of a vector).
[email protected] MATH 532 6 In particular, x, y are orthogonal if and only if xT y = 0.
Vector Norms
Remark Any inner product h·, ·i induces a norm via (more later) p kxk = hx, xi.
We will show that the standard inner product induces the Euclidean norm (cf. length of a vector).
Remark Inner products let us define angles via
xT y cos θ = . kxkkyk
[email protected] MATH 532 6 Vector Norms
Remark Any inner product h·, ·i induces a norm via (more later) p kxk = hx, xi.
We will show that the standard inner product induces the Euclidean norm (cf. length of a vector).
Remark Inner products let us define angles via
xT y cos θ = . kxkkyk
In particular, x, y are orthogonal if and only if xT y = 0.
[email protected] MATH 532 6 We show that k · k2 is a norm. We do this for the real case, but the complex case goes analogously.
1 Clearly, kxk2 ≥ 0. Also,
2 kxk2 = 0 ⇐⇒ kxk2 = 0 n X 2 ⇐⇒ xi = 0 ⇐⇒ xi = 0, i = 1,..., n, i=1 ⇐⇒ x = 0.
Vector Norms
Example Let x ∈ Rn and consider the Euclidean norm √ T kxk2 = x x n !1/2 X 2 = xi . i=1
[email protected] MATH 532 7 1 Clearly, kxk2 ≥ 0. Also,
2 kxk2 = 0 ⇐⇒ kxk2 = 0 n X 2 ⇐⇒ xi = 0 ⇐⇒ xi = 0, i = 1,..., n, i=1 ⇐⇒ x = 0.
Vector Norms
Example Let x ∈ Rn and consider the Euclidean norm √ T kxk2 = x x n !1/2 X 2 = xi . i=1
We show that k · k2 is a norm. We do this for the real case, but the complex case goes analogously.
[email protected] MATH 532 7 Vector Norms
Example Let x ∈ Rn and consider the Euclidean norm √ T kxk2 = x x n !1/2 X 2 = xi . i=1
We show that k · k2 is a norm. We do this for the real case, but the complex case goes analogously.
1 Clearly, kxk2 ≥ 0. Also,
2 kxk2 = 0 ⇐⇒ kxk2 = 0 n X 2 ⇐⇒ xi = 0 ⇐⇒ xi = 0, i = 1,..., n, i=1 ⇐⇒ x = 0.
[email protected] MATH 532 7 n !1/2 X 2 = |α| xi = |α| kxk2. i=1
3 To establish (3) we need Lemma Let x, y ∈ Rn. Then
T |x y| ≤ kxk2kyk2. (Cauchy–Schwarz–Bunyakovsky)
Moreover, equality holds if and only if y = αx with
xT y α = . 2 kxk2
Vector Norms Example (cont.) 2 We have
n !1/2 X 2 kαxk2 = (αxi ) i=1
[email protected] MATH 532 8 3 To establish (3) we need Lemma Let x, y ∈ Rn. Then
T |x y| ≤ kxk2kyk2. (Cauchy–Schwarz–Bunyakovsky)
Moreover, equality holds if and only if y = αx with
xT y α = . 2 kxk2
Vector Norms Example (cont.) 2 We have
n !1/2 n !1/2 X 2 X 2 kαxk2 = (αxi ) = |α| xi = |α| kxk2. i=1 i=1
[email protected] MATH 532 8 Vector Norms Example (cont.) 2 We have
n !1/2 n !1/2 X 2 X 2 kαxk2 = (αxi ) = |α| xi = |α| kxk2. i=1 i=1
3 To establish (3) we need Lemma Let x, y ∈ Rn. Then
T |x y| ≤ kxk2kyk2. (Cauchy–Schwarz–Bunyakovsky)
Moreover, equality holds if and only if y = αx with
xT y α = . 2 kxk2
[email protected] MATH 532 8 aT b cos θ = . kakkbk
Using trigonometry as in the figure, the projection of a onto b is then b aT b b aT b kak cos θ = kak = b. kbk kakkbk kbk kbk2
Now, we let y = a and x = b, so that the projection of y onto x is given by xT y αx, where α = . kxk2
Vector Norms Motivation for Proof of Cauchy–Schwarz–Bunyakovsky
As already alluded to above, the angle θ between two vectors a and b is related to the inner product by
[email protected] MATH 532 9 Using trigonometry as in the figure, the projection of a onto b is then b aT b b aT b kak cos θ = kak = b. kbk kakkbk kbk kbk2
Now, we let y = a and x = b, so that the projection of y onto x is given by xT y αx, where α = . kxk2
Vector Norms Motivation for Proof of Cauchy–Schwarz–Bunyakovsky
As already alluded to above, the angle θ between two vectors a and b is related to the inner product by
aT b cos θ = . kakkbk
[email protected] MATH 532 9 the projection of a onto b is then b aT b b aT b kak cos θ = kak = b. kbk kakkbk kbk kbk2
Now, we let y = a and x = b, so that the projection of y onto x is given by xT y αx, where α = . kxk2
Vector Norms Motivation for Proof of Cauchy–Schwarz–Bunyakovsky
As already alluded to above, the angle θ between two vectors a and b is related to the inner product by
aT b cos θ = . kakkbk
Using trigonometry as in the figure,
[email protected] MATH 532 9 aT b b aT b = kak = b. kakkbk kbk kbk2
Now, we let y = a and x = b, so that the projection of y onto x is given by xT y αx, where α = . kxk2
Vector Norms Motivation for Proof of Cauchy–Schwarz–Bunyakovsky
As already alluded to above, the angle θ between two vectors a and b is related to the inner product by
aT b cos θ = . kakkbk
Using trigonometry as in the figure, the projection of a onto b is then b kak cos θ kbk
[email protected] MATH 532 9 aT b = b. kbk2
Now, we let y = a and x = b, so that the projection of y onto x is given by xT y αx, where α = . kxk2
Vector Norms Motivation for Proof of Cauchy–Schwarz–Bunyakovsky
As already alluded to above, the angle θ between two vectors a and b is related to the inner product by
aT b cos θ = . kakkbk
Using trigonometry as in the figure, the projection of a onto b is then b aT b b kak cos θ = kak kbk kakkbk kbk
[email protected] MATH 532 9 Now, we let y = a and x = b, so that the projection of y onto x is given by xT y αx, where α = . kxk2
Vector Norms Motivation for Proof of Cauchy–Schwarz–Bunyakovsky
As already alluded to above, the angle θ between two vectors a and b is related to the inner product by
aT b cos θ = . kakkbk
Using trigonometry as in the figure, the projection of a onto b is then b aT b b aT b kak cos θ = kak = b. kbk kakkbk kbk kbk2
[email protected] MATH 532 9 Vector Norms Motivation for Proof of Cauchy–Schwarz–Bunyakovsky
As already alluded to above, the angle θ between two vectors a and b is related to the inner product by
aT b cos θ = . kakkbk
Using trigonometry as in the figure, the projection of a onto b is then b aT b b aT b kak cos θ = kak = b. kbk kakkbk kbk kbk2
Now, we let y = a and x = b, so that the projection of y onto x is given by xT y αx, where α = . kxk2
[email protected] MATH 532 9 = y T y − 2αxT y + α2xT x 2 xT y xT y = y T y − 2 xT y + xT x kxk2 kxk4 |{z} 2 =kxk2 2 xT y = k k2 − . y 2 2 kxk2 This implies 2 T 2 2 x y ≤ kxk2kyk2, and the Cauchy–Schwarz–Bunyakovsky inequality follows by taking square roots.
since it’s (the square of) a norm. Therefore,
2 T 0 ≤ ky − αxk2 = (y − αx) (y − αx)
Vector Norms Proof of Cauchy–Schwarz–Bunyakovsky 2 We know that ky − αxk2 ≥ 0
[email protected] MATH 532 10 = y T y − 2αxT y + α2xT x 2 xT y xT y = y T y − 2 xT y + xT x kxk2 kxk4 |{z} 2 =kxk2 2 xT y = k k2 − . y 2 2 kxk2 This implies 2 T 2 2 x y ≤ kxk2kyk2, and the Cauchy–Schwarz–Bunyakovsky inequality follows by taking square roots.
Therefore,
2 T 0 ≤ ky − αxk2 = (y − αx) (y − αx)
Vector Norms Proof of Cauchy–Schwarz–Bunyakovsky 2 We know that ky − αxk2 ≥ 0 since it’s (the square of) a norm.
[email protected] MATH 532 10 = y T y − 2αxT y + α2xT x 2 xT y xT y = y T y − 2 xT y + xT x kxk2 kxk4 |{z} 2 =kxk2 2 xT y = k k2 − . y 2 2 kxk2 This implies 2 T 2 2 x y ≤ kxk2kyk2, and the Cauchy–Schwarz–Bunyakovsky inequality follows by taking square roots.
Vector Norms Proof of Cauchy–Schwarz–Bunyakovsky 2 We know that ky − αxk2 ≥ 0 since it’s (the square of) a norm. Therefore,
2 T 0 ≤ ky − αxk2 = (y − αx) (y − αx)
[email protected] MATH 532 10 2 xT y xT y = y T y − 2 xT y + xT x kxk2 kxk4 |{z} 2 =kxk2 2 xT y = k k2 − . y 2 2 kxk2 This implies 2 T 2 2 x y ≤ kxk2kyk2, and the Cauchy–Schwarz–Bunyakovsky inequality follows by taking square roots.
Vector Norms Proof of Cauchy–Schwarz–Bunyakovsky 2 We know that ky − αxk2 ≥ 0 since it’s (the square of) a norm. Therefore,
2 T 0 ≤ ky − αxk2 = (y − αx) (y − αx) = y T y − 2αxT y + α2xT x
[email protected] MATH 532 10 2 xT y = k k2 − . y 2 2 kxk2 This implies 2 T 2 2 x y ≤ kxk2kyk2, and the Cauchy–Schwarz–Bunyakovsky inequality follows by taking square roots.
Vector Norms Proof of Cauchy–Schwarz–Bunyakovsky 2 We know that ky − αxk2 ≥ 0 since it’s (the square of) a norm. Therefore,
2 T 0 ≤ ky − αxk2 = (y − αx) (y − αx) = y T y − 2αxT y + α2xT x 2 xT y xT y = y T y − 2 xT y + xT x kxk2 kxk4 |{z} 2 =kxk2
[email protected] MATH 532 10 This implies 2 T 2 2 x y ≤ kxk2kyk2, and the Cauchy–Schwarz–Bunyakovsky inequality follows by taking square roots.
Vector Norms Proof of Cauchy–Schwarz–Bunyakovsky 2 We know that ky − αxk2 ≥ 0 since it’s (the square of) a norm. Therefore,
2 T 0 ≤ ky − αxk2 = (y − αx) (y − αx) = y T y − 2αxT y + α2xT x 2 xT y xT y = y T y − 2 xT y + xT x kxk2 kxk4 |{z} 2 =kxk2 2 xT y = k k2 − . y 2 2 kxk2
[email protected] MATH 532 10 and the Cauchy–Schwarz–Bunyakovsky inequality follows by taking square roots.
Vector Norms Proof of Cauchy–Schwarz–Bunyakovsky 2 We know that ky − αxk2 ≥ 0 since it’s (the square of) a norm. Therefore,
2 T 0 ≤ ky − αxk2 = (y − αx) (y − αx) = y T y − 2αxT y + α2xT x 2 xT y xT y = y T y − 2 xT y + xT x kxk2 kxk4 |{z} 2 =kxk2 2 xT y = k k2 − . y 2 2 kxk2 This implies 2 T 2 2 x y ≤ kxk2kyk2,
[email protected] MATH 532 10 Vector Norms Proof of Cauchy–Schwarz–Bunyakovsky 2 We know that ky − αxk2 ≥ 0 since it’s (the square of) a norm. Therefore,
2 T 0 ≤ ky − αxk2 = (y − αx) (y − αx) = y T y − 2αxT y + α2xT x 2 xT y xT y = y T y − 2 xT y + xT x kxk2 kxk4 |{z} 2 =kxk2 2 xT y = k k2 − . y 2 2 kxk2 This implies 2 T 2 2 x y ≤ kxk2kyk2, and the Cauchy–Schwarz–Bunyakovsky inequality follows by taking square roots. [email protected] MATH 532 10 T “=⇒”: Let’s assume that x y = kxk2kyk2. But then the first part of the proof shows that ky − αxk2 = 0 so that y = αx.
“⇐=”: Let’s assume y = αx. Then
T T 2 x y = x (αx) = |α|kxk2 2 kxk2kyk2 = kxk2kαxk2 = |α|kxk2,
so that we have equality.
Vector Norms
Proof (cont.) Now we look at the equality claim.
[email protected] MATH 532 11 But then the first part of the proof shows that ky − αxk2 = 0 so that y = αx.
“⇐=”: Let’s assume y = αx. Then
T T 2 x y = x (αx) = |α|kxk2 2 kxk2kyk2 = kxk2kαxk2 = |α|kxk2,
so that we have equality.
Vector Norms
Proof (cont.) Now we look at the equality claim.
T “=⇒”: Let’s assume that x y = kxk2kyk2.
[email protected] MATH 532 11 “⇐=”: Let’s assume y = αx. Then
T T 2 x y = x (αx) = |α|kxk2 2 kxk2kyk2 = kxk2kαxk2 = |α|kxk2,
so that we have equality.
Vector Norms
Proof (cont.) Now we look at the equality claim.
T “=⇒”: Let’s assume that x y = kxk2kyk2. But then the first part of the proof shows that ky − αxk2 = 0 so that y = αx.
[email protected] MATH 532 11 Then
T T 2 x y = x (αx) = |α|kxk2 2 kxk2kyk2 = kxk2kαxk2 = |α|kxk2,
so that we have equality.
Vector Norms
Proof (cont.) Now we look at the equality claim.
T “=⇒”: Let’s assume that x y = kxk2kyk2. But then the first part of the proof shows that ky − αxk2 = 0 so that y = αx.
“⇐=”: Let’s assume y = αx.
[email protected] MATH 532 11 Vector Norms
Proof (cont.) Now we look at the equality claim.
T “=⇒”: Let’s assume that x y = kxk2kyk2. But then the first part of the proof shows that ky − αxk2 = 0 so that y = αx.
“⇐=”: Let’s assume y = αx. Then
T T 2 x y = x (αx) = |α|kxk2 2 kxk2kyk2 = kxk2kαxk2 = |α|kxk2, so that we have equality.
[email protected] MATH 532 11 2 T kx + yk2 = (x + y) (x + y) = xT x +xT y + y T x + y T y |{z} |{z} |{z} =kxk2 = T 2 2 x y =kyk2 2 T 2 = kxk2 + 2x y + kyk2 2 T 2 ≤ kxk2 + 2|x y| + kyk2 CSB 2 2 ≤ kxk2 + 2kxk2kyk2 + kyk2 2 = (kxk2 + kyk2) .
Now we just need to take square roots to have the triangle inequality.
Vector Norms
Example (cont.)
3 Now we can show that k · k2 satisfies the triangle inequality:
[email protected] MATH 532 12 = xT x +xT y + y T x + y T y |{z} |{z} |{z} =kxk2 = T 2 2 x y =kyk2 2 T 2 = kxk2 + 2x y + kyk2 2 T 2 ≤ kxk2 + 2|x y| + kyk2 CSB 2 2 ≤ kxk2 + 2kxk2kyk2 + kyk2 2 = (kxk2 + kyk2) .
Now we just need to take square roots to have the triangle inequality.
Vector Norms
Example (cont.)
3 Now we can show that k · k2 satisfies the triangle inequality:
2 T kx + yk2 = (x + y) (x + y)
[email protected] MATH 532 12 2 T 2 = kxk2 + 2x y + kyk2 2 T 2 ≤ kxk2 + 2|x y| + kyk2 CSB 2 2 ≤ kxk2 + 2kxk2kyk2 + kyk2 2 = (kxk2 + kyk2) .
Now we just need to take square roots to have the triangle inequality.
Vector Norms
Example (cont.)
3 Now we can show that k · k2 satisfies the triangle inequality:
2 T kx + yk2 = (x + y) (x + y) = xT x +xT y + y T x + y T y |{z} |{z} |{z} =kxk2 = T 2 2 x y =kyk2
[email protected] MATH 532 12 2 T 2 ≤ kxk2 + 2|x y| + kyk2 CSB 2 2 ≤ kxk2 + 2kxk2kyk2 + kyk2 2 = (kxk2 + kyk2) .
Now we just need to take square roots to have the triangle inequality.
Vector Norms
Example (cont.)
3 Now we can show that k · k2 satisfies the triangle inequality:
2 T kx + yk2 = (x + y) (x + y) = xT x +xT y + y T x + y T y |{z} |{z} |{z} =kxk2 = T 2 2 x y =kyk2 2 T 2 = kxk2 + 2x y + kyk2
[email protected] MATH 532 12 CSB 2 2 ≤ kxk2 + 2kxk2kyk2 + kyk2 2 = (kxk2 + kyk2) .
Now we just need to take square roots to have the triangle inequality.
Vector Norms
Example (cont.)
3 Now we can show that k · k2 satisfies the triangle inequality:
2 T kx + yk2 = (x + y) (x + y) = xT x +xT y + y T x + y T y |{z} |{z} |{z} =kxk2 = T 2 2 x y =kyk2 2 T 2 = kxk2 + 2x y + kyk2 2 T 2 ≤ kxk2 + 2|x y| + kyk2
[email protected] MATH 532 12 2 = (kxk2 + kyk2) .
Now we just need to take square roots to have the triangle inequality.
Vector Norms
Example (cont.)
3 Now we can show that k · k2 satisfies the triangle inequality:
2 T kx + yk2 = (x + y) (x + y) = xT x +xT y + y T x + y T y |{z} |{z} |{z} =kxk2 = T 2 2 x y =kyk2 2 T 2 = kxk2 + 2x y + kyk2 2 T 2 ≤ kxk2 + 2|x y| + kyk2 CSB 2 2 ≤ kxk2 + 2kxk2kyk2 + kyk2
[email protected] MATH 532 12 Now we just need to take square roots to have the triangle inequality.
Vector Norms
Example (cont.)
3 Now we can show that k · k2 satisfies the triangle inequality:
2 T kx + yk2 = (x + y) (x + y) = xT x +xT y + y T x + y T y |{z} |{z} |{z} =kxk2 = T 2 2 x y =kyk2 2 T 2 = kxk2 + 2x y + kyk2 2 T 2 ≤ kxk2 + 2|x y| + kyk2 CSB 2 2 ≤ kxk2 + 2kxk2kyk2 + kyk2 2 = (kxk2 + kyk2) .
[email protected] MATH 532 12 Vector Norms
Example (cont.)
3 Now we can show that k · k2 satisfies the triangle inequality:
2 T kx + yk2 = (x + y) (x + y) = xT x +xT y + y T x + y T y |{z} |{z} |{z} =kxk2 = T 2 2 x y =kyk2 2 T 2 = kxk2 + 2x y + kyk2 2 T 2 ≤ kxk2 + 2|x y| + kyk2 CSB 2 2 ≤ kxk2 + 2kxk2kyk2 + kyk2 2 = (kxk2 + kyk2) .
Now we just need to take square roots to have the triangle inequality.
[email protected] MATH 532 12 Proof We write tri.ineq. kxk = kx − y + yk ≤ kx − yk + kyk. But this implies kxk − kyk ≤ kx − yk.
Vector Norms
Lemma Let x, y ∈ Rn. Then we have the backward triangle inequality
| kxk − kyk | ≤ kx − yk.
[email protected] MATH 532 13 tri.ineq. ≤ kx − yk + kyk. But this implies kxk − kyk ≤ kx − yk.
Vector Norms
Lemma Let x, y ∈ Rn. Then we have the backward triangle inequality
| kxk − kyk | ≤ kx − yk.
Proof We write kxk = kx − y + yk
[email protected] MATH 532 13 But this implies kxk − kyk ≤ kx − yk.
Vector Norms
Lemma Let x, y ∈ Rn. Then we have the backward triangle inequality
| kxk − kyk | ≤ kx − yk.
Proof We write tri.ineq. kxk = kx − y + yk ≤ kx − yk + kyk.
[email protected] MATH 532 13 Vector Norms
Lemma Let x, y ∈ Rn. Then we have the backward triangle inequality
| kxk − kyk | ≤ kx − yk.
Proof We write tri.ineq. kxk = kx − y + yk ≤ kx − yk + kyk. But this implies kxk − kyk ≤ kx − yk.
[email protected] MATH 532 13 ⇐⇒ − (kxk − kyk) ≤ kx − yk.
Together with the previous inequality we have
|kxk − kyk| ≤ kx − yk.
Vector Norms
Proof (cont.) Switch the roles of x and y to get
kyk − kxk ≤ ky − xk
[email protected] MATH 532 14 Together with the previous inequality we have
|kxk − kyk| ≤ kx − yk.
Vector Norms
Proof (cont.) Switch the roles of x and y to get
kyk − kxk ≤ ky − xk ⇐⇒ − (kxk − kyk) ≤ kx − yk.
[email protected] MATH 532 14 Vector Norms
Proof (cont.) Switch the roles of x and y to get
kyk − kxk ≤ ky − xk ⇐⇒ − (kxk − kyk) ≤ kx − yk.
Together with the previous inequality we have
|kxk − kyk| ≤ kx − yk.
[email protected] MATH 532 14 `1-norm (or taxi-cab norm, Manhattan norm): n X kxk1 = |xi | i=1
`∞-norm (or maximum norm, Chebyshev norm):
kxk∞ = max |xi | 1≤i≤n
`p-norm: n !1/p X p kxkp = |xi | i=1
Remark In the homework you will use Hölder’s and Minkowski’s inequalities to show that the p-norm is a norm.
Vector Norms Other common norms
[email protected] MATH 532 15 `∞-norm (or maximum norm, Chebyshev norm):
kxk∞ = max |xi | 1≤i≤n
`p-norm: n !1/p X p kxkp = |xi | i=1
Remark In the homework you will use Hölder’s and Minkowski’s inequalities to show that the p-norm is a norm.
Vector Norms Other common norms
`1-norm (or taxi-cab norm, Manhattan norm): n X kxk1 = |xi | i=1
[email protected] MATH 532 15 `p-norm: n !1/p X p kxkp = |xi | i=1
Remark In the homework you will use Hölder’s and Minkowski’s inequalities to show that the p-norm is a norm.
Vector Norms Other common norms
`1-norm (or taxi-cab norm, Manhattan norm): n X kxk1 = |xi | i=1
`∞-norm (or maximum norm, Chebyshev norm):
kxk∞ = max |xi | 1≤i≤n
[email protected] MATH 532 15 Remark In the homework you will use Hölder’s and Minkowski’s inequalities to show that the p-norm is a norm.
Vector Norms Other common norms
`1-norm (or taxi-cab norm, Manhattan norm): n X kxk1 = |xi | i=1
`∞-norm (or maximum norm, Chebyshev norm):
kxk∞ = max |xi | 1≤i≤n
`p-norm: n !1/p X p kxkp = |xi | i=1
[email protected] MATH 532 15 Vector Norms Other common norms
`1-norm (or taxi-cab norm, Manhattan norm): n X kxk1 = |xi | i=1
`∞-norm (or maximum norm, Chebyshev norm):
kxk∞ = max |xi | 1≤i≤n
`p-norm: n !1/p X p kxkp = |xi | i=1
Remark In the homework you will use Hölder’s and Minkowski’s inequalities to show that the p-norm is a norm. [email protected] MATH 532 15 Let’s use tildes to mark all components of x that are maximal, i.e..
x˜1 = x˜2 = ... = x˜k = max |xi |. 1≤i≤n
The remaining components are then x˜k+1,..., x˜n. This implies that
x˜ i < 1, for i = k + 1,..., n. x˜1
Vector Norms
Remark We now show that kxk∞ = lim kxkp. p→∞
[email protected] MATH 532 16 The remaining components are then x˜k+1,..., x˜n. This implies that
x˜ i < 1, for i = k + 1,..., n. x˜1
Vector Norms
Remark We now show that kxk∞ = lim kxkp. p→∞ Let’s use tildes to mark all components of x that are maximal, i.e..
x˜1 = x˜2 = ... = x˜k = max |xi |. 1≤i≤n
[email protected] MATH 532 16 This implies that
x˜ i < 1, for i = k + 1,..., n. x˜1
Vector Norms
Remark We now show that kxk∞ = lim kxkp. p→∞ Let’s use tildes to mark all components of x that are maximal, i.e..
x˜1 = x˜2 = ... = x˜k = max |xi |. 1≤i≤n
The remaining components are then x˜k+1,..., x˜n.
[email protected] MATH 532 16 Vector Norms
Remark We now show that kxk∞ = lim kxkp. p→∞ Let’s use tildes to mark all components of x that are maximal, i.e..
x˜1 = x˜2 = ... = x˜k = max |xi |. 1≤i≤n
The remaining components are then x˜k+1,..., x˜n. This implies that
x˜ i < 1, for i = k + 1,..., n. x˜1
[email protected] MATH 532 16 1/p x˜ p x˜ p ˜ k+1 n = |x1| k + + ... + . x˜1 x˜1 | {z } |{z} <1 <1
Since the terms inside the parentheses — except for k — go to 0 for p → ∞, (·)1/p → 1 for p → ∞. And so kxkp → |x˜1| = max |xi | = kxk∞. 1≤i≤n
Vector Norms
Remark (cont.) Now
n !1/p X p kxkp = |x˜i | i=1
[email protected] MATH 532 17 Since the terms inside the parentheses — except for k — go to 0 for p → ∞, (·)1/p → 1 for p → ∞. And so kxkp → |x˜1| = max |xi | = kxk∞. 1≤i≤n
Vector Norms
Remark (cont.) Now
n !1/p X p kxkp = |x˜i | i=1 1/p x˜ p x˜ p ˜ k+1 n = |x1| k + + ... + . x˜1 x˜1 | {z } |{z} <1 <1
[email protected] MATH 532 17 And so kxkp → |x˜1| = max |xi | = kxk∞. 1≤i≤n
Vector Norms
Remark (cont.) Now
n !1/p X p kxkp = |x˜i | i=1 1/p x˜ p x˜ p ˜ k+1 n = |x1| k + + ... + . x˜1 x˜1 | {z } |{z} <1 <1
Since the terms inside the parentheses — except for k — go to 0 for p → ∞, (·)1/p → 1 for p → ∞.
[email protected] MATH 532 17 Vector Norms
Remark (cont.) Now
n !1/p X p kxkp = |x˜i | i=1 1/p x˜ p x˜ p ˜ k+1 n = |x1| k + + ... + . x˜1 x˜1 | {z } |{z} <1 <1
Since the terms inside the parentheses — except for k — go to 0 for p → ∞, (·)1/p → 1 for p → ∞. And so kxkp → |x˜1| = max |xi | = kxk∞. 1≤i≤n
[email protected] MATH 532 17 √ ! √ ! √ ! 2 2 2 so that √2 ≥ √2 ≥ √2 . 2 2 2 2 1 2 2 2 ∞ In fact, we have in general (similar to HW) n kxk1 ≥ kxk2 ≥ kxk∞, for any x ∈ R .
Vector Norms
2 Figure: Unit “balls” in R for the `1, `2 and `∞ norms.
Note that B1 ⊆ B2 ⊆ B∞ since, e.g., √ ! √ ! √ ! √ 2 √ 2 q 2 2 √2 = 2, √2 = 1 + 1 = 1, √2 = , 2 2 2 2 2 2 2 1 2 2 2 ∞
[email protected] MATH 532 18 In fact, we have in general (similar to HW) n kxk1 ≥ kxk2 ≥ kxk∞, for any x ∈ R .
Vector Norms
2 Figure: Unit “balls” in R for the `1, `2 and `∞ norms.
Note that B1 ⊆ B2 ⊆ B∞ since, e.g., √ ! √ ! √ ! √ 2 √ 2 q 2 2 √2 = 2, √2 = 1 + 1 = 1, √2 = , 2 2 2 2 2 2 2 1 2 2 2 ∞ √ ! √ ! √ ! 2 2 2 so that √2 ≥ √2 ≥ √2 . 2 2 2 2 1 2 2 2 ∞
[email protected] MATH 532 18 Vector Norms
2 Figure: Unit “balls” in R for the `1, `2 and `∞ norms.
Note that B1 ⊆ B2 ⊆ B∞ since, e.g., √ ! √ ! √ ! √ 2 √ 2 q 2 2 √2 = 2, √2 = 1 + 1 = 1, √2 = , 2 2 2 2 2 2 2 1 2 2 2 ∞ √ ! √ ! √ ! 2 2 2 so that √2 ≥ √2 ≥ √2 . 2 2 2 2 1 2 2 2 ∞ In fact, we have in general (similar to HW) n kxk1 ≥ kxk2 ≥ kxk∞, for any x ∈ R .
[email protected] MATH 532 18 Example
k · k1 and√ k · k2 are equivalent since from above kxk1 ≥ kxk2 and also kxk1 ≤ nkxk2 (see HW) so that kxk √ α = 1 ≤ 1 ≤ n = β. kxk2
Remark In fact, all norms on finite-dimensional vector spaces are equivalent.
Vector Norms Norm equivalence Definition Two norms k · k and k · k0 on a vector space V are called equivalent if there exist constants α, β such that
kxk α ≤ ≤ β for all x(6= 0) ∈ V. kxk0
[email protected] MATH 532 19 kxk √ α = 1 ≤ 1 ≤ n = β. kxk2
Remark In fact, all norms on finite-dimensional vector spaces are equivalent.
Vector Norms Norm equivalence Definition Two norms k · k and k · k0 on a vector space V are called equivalent if there exist constants α, β such that
kxk α ≤ ≤ β for all x(6= 0) ∈ V. kxk0
Example
k · k1 and√ k · k2 are equivalent since from above kxk1 ≥ kxk2 and also kxk1 ≤ nkxk2 (see HW) so that
[email protected] MATH 532 19 Remark In fact, all norms on finite-dimensional vector spaces are equivalent.
Vector Norms Norm equivalence Definition Two norms k · k and k · k0 on a vector space V are called equivalent if there exist constants α, β such that
kxk α ≤ ≤ β for all x(6= 0) ∈ V. kxk0
Example
k · k1 and√ k · k2 are equivalent since from above kxk1 ≥ kxk2 and also kxk1 ≤ nkxk2 (see HW) so that kxk √ α = 1 ≤ 1 ≤ n = β. kxk2
[email protected] MATH 532 19 Vector Norms Norm equivalence Definition Two norms k · k and k · k0 on a vector space V are called equivalent if there exist constants α, β such that
kxk α ≤ ≤ β for all x(6= 0) ∈ V. kxk0
Example
k · k1 and√ k · k2 are equivalent since from above kxk1 ≥ kxk2 and also kxk1 ≤ nkxk2 (see HW) so that kxk √ α = 1 ≤ 1 ≤ n = β. kxk2
Remark In fact, all norms on finite-dimensional vector spaces are equivalent.
[email protected] MATH 532 19 Matrix Norms Outline
1 Vector Norms
2 Matrix Norms
3 Inner Product Spaces
4 Orthogonal Vectors
5 Gram–Schmidt Orthogonalization & QR Factorization
6 Unitary and Orthogonal Matrices
7 Orthogonal Reduction
8 Complementary Subspaces
9 Orthogonal Decomposition
10 Singular Value Decomposition
11 Orthogonal Projections
[email protected] MATH 532 20 1/2 m n m !1/2 X X 2 X 2 kAkF = |aij | = kAi∗k2 i=1 j=1 i=1 1/2 n q X 2 T = kA∗j k2 = trace(A A), j=1
i.e., the Frobenius norm is just a 2-norm for the vector that contains all elements of the matrix.
This property should help us measure kABk for two matrices A, B of appropriate sizes. We look at the simplest matrix norm, the Frobenius norm, defined for , A ∈ Rm n:
Matrix Norms
Matrix norms are special norms — they will satisfy one additional property.
[email protected] MATH 532 21 1/2 m n m !1/2 X X 2 X 2 kAkF = |aij | = kAi∗k2 i=1 j=1 i=1 1/2 n q X 2 T = kA∗j k2 = trace(A A), j=1
i.e., the Frobenius norm is just a 2-norm for the vector that contains all elements of the matrix.
We look at the simplest matrix norm, the Frobenius norm, defined for , A ∈ Rm n:
Matrix Norms
Matrix norms are special norms — they will satisfy one additional property. This property should help us measure kABk for two matrices A, B of appropriate sizes.
[email protected] MATH 532 21 1/2 m n m !1/2 X X 2 X 2 kAkF = |aij | = kAi∗k2 i=1 j=1 i=1 1/2 n q X 2 T = kA∗j k2 = trace(A A), j=1
i.e., the Frobenius norm is just a 2-norm for the vector that contains all elements of the matrix.
Matrix Norms
Matrix norms are special norms — they will satisfy one additional property. This property should help us measure kABk for two matrices A, B of appropriate sizes. We look at the simplest matrix norm, the Frobenius norm, defined for , A ∈ Rm n:
[email protected] MATH 532 21 m !1/2 X 2 = kAi∗k2 i=1 1/2 n q X 2 T = kA∗j k2 = trace(A A), j=1
i.e., the Frobenius norm is just a 2-norm for the vector that contains all elements of the matrix.
Matrix Norms
Matrix norms are special norms — they will satisfy one additional property. This property should help us measure kABk for two matrices A, B of appropriate sizes. We look at the simplest matrix norm, the Frobenius norm, defined for , A ∈ Rm n: 1/2 m n X X 2 kAkF = |aij | i=1 j=1
[email protected] MATH 532 21 1/2 n q X 2 T = kA∗j k2 = trace(A A), j=1
i.e., the Frobenius norm is just a 2-norm for the vector that contains all elements of the matrix.
Matrix Norms
Matrix norms are special norms — they will satisfy one additional property. This property should help us measure kABk for two matrices A, B of appropriate sizes. We look at the simplest matrix norm, the Frobenius norm, defined for , A ∈ Rm n: 1/2 m n m !1/2 X X 2 X 2 kAkF = |aij | = kAi∗k2 i=1 j=1 i=1
[email protected] MATH 532 21 q = trace(AT A),
i.e., the Frobenius norm is just a 2-norm for the vector that contains all elements of the matrix.
Matrix Norms
Matrix norms are special norms — they will satisfy one additional property. This property should help us measure kABk for two matrices A, B of appropriate sizes. We look at the simplest matrix norm, the Frobenius norm, defined for , A ∈ Rm n: 1/2 m n m !1/2 X X 2 X 2 kAkF = |aij | = kAi∗k2 i=1 j=1 i=1 1/2 n X 2 = kA∗j k2 j=1
[email protected] MATH 532 21 i.e., the Frobenius norm is just a 2-norm for the vector that contains all elements of the matrix.
Matrix Norms
Matrix norms are special norms — they will satisfy one additional property. This property should help us measure kABk for two matrices A, B of appropriate sizes. We look at the simplest matrix norm, the Frobenius norm, defined for , A ∈ Rm n: 1/2 m n m !1/2 X X 2 X 2 kAkF = |aij | = kAi∗k2 i=1 j=1 i=1 1/2 n q X 2 T = kA∗j k2 = trace(A A), j=1
[email protected] MATH 532 21 Matrix Norms
Matrix norms are special norms — they will satisfy one additional property. This property should help us measure kABk for two matrices A, B of appropriate sizes. We look at the simplest matrix norm, the Frobenius norm, defined for , A ∈ Rm n: 1/2 m n m !1/2 X X 2 X 2 kAkF = |aij | = kAi∗k2 i=1 j=1 i=1 1/2 n q X 2 T = kA∗j k2 = trace(A A), j=1 i.e., the Frobenius norm is just a 2-norm for the vector that contains all elements of the matrix.
[email protected] MATH 532 21 m CSB X 2 2 ≤ kAi∗k2 kxk2 i=1 | {z } 2 =kAkF
so that kAxk2 ≤ kAkF kxk2.
We can generalize this to matrices, i.e., we have
kABkF ≤ kAkF kBkF ,
which motivates us to require this submultiplicativity for any matrix norm.
Matrix Norms Now
m 2 X 2 kAxk2 = |Ai∗x| i=1
[email protected] MATH 532 22 so that kAxk2 ≤ kAkF kxk2.
We can generalize this to matrices, i.e., we have
kABkF ≤ kAkF kBkF ,
which motivates us to require this submultiplicativity for any matrix norm.
Matrix Norms Now
m 2 X 2 kAxk2 = |Ai∗x| i=1 m CSB X 2 2 ≤ kAi∗k2 kxk2 i=1 | {z } 2 =kAkF
[email protected] MATH 532 22 We can generalize this to matrices, i.e., we have
kABkF ≤ kAkF kBkF ,
which motivates us to require this submultiplicativity for any matrix norm.
Matrix Norms Now
m 2 X 2 kAxk2 = |Ai∗x| i=1 m CSB X 2 2 ≤ kAi∗k2 kxk2 i=1 | {z } 2 =kAkF so that kAxk2 ≤ kAkF kxk2.
[email protected] MATH 532 22 Matrix Norms Now
m 2 X 2 kAxk2 = |Ai∗x| i=1 m CSB X 2 2 ≤ kAi∗k2 kxk2 i=1 | {z } 2 =kAkF so that kAxk2 ≤ kAkF kxk2.
We can generalize this to matrices, i.e., we have
kABkF ≤ kAkF kBkF , which motivates us to require this submultiplicativity for any matrix norm. [email protected] MATH 532 22 Remark This definition is usually too general. In addition to the Frobenius norm, most useful matrix norms are induced by a vector norm.
Matrix Norms
Definition A matrix norm is a function k · k from the set of all real (or complex) matrices of finite size into R≥0 that satisfies 1 kAk ≥ 0 and kAk = 0 if and only if A = O (a matrix of all zeros). 2 kαAk = |α|kAk for all α ∈ R. 3 kA + Bk ≤ kAk + kBk (requires A, B to be of same size). 4 kABk ≤ kAkkBk (requires A, B to have appropriate sizes).
[email protected] MATH 532 23 Matrix Norms
Definition A matrix norm is a function k · k from the set of all real (or complex) matrices of finite size into R≥0 that satisfies 1 kAk ≥ 0 and kAk = 0 if and only if A = O (a matrix of all zeros). 2 kαAk = |α|kAk for all α ∈ R. 3 kA + Bk ≤ kAk + kBk (requires A, B to be of same size). 4 kABk ≤ kAkkBk (requires A, B to have appropriate sizes).
Remark This definition is usually too general. In addition to the Frobenius norm, most useful matrix norms are induced by a vector norm.
[email protected] MATH 532 23 Remark Here the vector norm could be any vector norm. In particular, any p-norm. For example, we could have
kAk2 = max kAxk2,(m). kxk2,(n)=1
To keep notation simple we often drop indices.
Matrix Norms Induced matrix norms
Theorem m n Let k · k(m) and k · k(n) be vector norms on R and R , respectively, and let A be an m × n matrix. Then
kAk = max kAxk(m) kxk(n)=1
is a matrix norm called the induced matrix norm.
[email protected] MATH 532 24 Matrix Norms Induced matrix norms
Theorem m n Let k · k(m) and k · k(n) be vector norms on R and R , respectively, and let A be an m × n matrix. Then
kAk = max kAxk(m) kxk(n)=1
is a matrix norm called the induced matrix norm.
Remark Here the vector norm could be any vector norm. In particular, any p-norm. For example, we could have
kAk2 = max kAxk2,(m). kxk2,(n)=1
To keep notation simple we often drop indices.
[email protected] MATH 532 24 It remains to show that kAk = 0 if and only if A = O. Assume A = O, then
kAk = max k Ax k = 0. kxk=1 |{z} =0
So now consider A 6= O. We need to show that kAk > 0. There must exist a column of A that is not 0. We call this column A∗k and take x = ek . Then kek k=1 kAk = max kAxk ≥ kAek k = kA∗k k > 0 kxk=1
since A∗k 6= 0.
Matrix Norms
Proof 1 kAk ≥ 0 is obvious since this holds for the vector norm.
[email protected] MATH 532 25 Assume A = O, then
kAk = max k Ax k = 0. kxk=1 |{z} =0
So now consider A 6= O. We need to show that kAk > 0. There must exist a column of A that is not 0. We call this column A∗k and take x = ek . Then kek k=1 kAk = max kAxk ≥ kAek k = kA∗k k > 0 kxk=1
since A∗k 6= 0.
Matrix Norms
Proof 1 kAk ≥ 0 is obvious since this holds for the vector norm. It remains to show that kAk = 0 if and only if A = O.
[email protected] MATH 532 25 So now consider A 6= O. We need to show that kAk > 0. There must exist a column of A that is not 0. We call this column A∗k and take x = ek . Then kek k=1 kAk = max kAxk ≥ kAek k = kA∗k k > 0 kxk=1
since A∗k 6= 0.
Matrix Norms
Proof 1 kAk ≥ 0 is obvious since this holds for the vector norm. It remains to show that kAk = 0 if and only if A = O. Assume A = O, then
kAk = max k Ax k = 0. kxk=1 |{z} =0
[email protected] MATH 532 25 There must exist a column of A that is not 0. We call this column A∗k and take x = ek . Then kek k=1 kAk = max kAxk ≥ kAek k = kA∗k k > 0 kxk=1
since A∗k 6= 0.
Matrix Norms
Proof 1 kAk ≥ 0 is obvious since this holds for the vector norm. It remains to show that kAk = 0 if and only if A = O. Assume A = O, then
kAk = max k Ax k = 0. kxk=1 |{z} =0
So now consider A 6= O. We need to show that kAk > 0.
[email protected] MATH 532 25 Then kek k=1 kAk = max kAxk ≥ kAek k = kA∗k k > 0 kxk=1
since A∗k 6= 0.
Matrix Norms
Proof 1 kAk ≥ 0 is obvious since this holds for the vector norm. It remains to show that kAk = 0 if and only if A = O. Assume A = O, then
kAk = max k Ax k = 0. kxk=1 |{z} =0
So now consider A 6= O. We need to show that kAk > 0. There must exist a column of A that is not 0. We call this column A∗k and take x = ek .
[email protected] MATH 532 25 kek k=1 ≥ kAek k = kA∗k k > 0
since A∗k 6= 0.
Matrix Norms
Proof 1 kAk ≥ 0 is obvious since this holds for the vector norm. It remains to show that kAk = 0 if and only if A = O. Assume A = O, then
kAk = max k Ax k = 0. kxk=1 |{z} =0
So now consider A 6= O. We need to show that kAk > 0. There must exist a column of A that is not 0. We call this column A∗k and take x = ek . Then kAk = max kAxk kxk=1
[email protected] MATH 532 25 = kA∗k k > 0
since A∗k 6= 0.
Matrix Norms
Proof 1 kAk ≥ 0 is obvious since this holds for the vector norm. It remains to show that kAk = 0 if and only if A = O. Assume A = O, then
kAk = max k Ax k = 0. kxk=1 |{z} =0
So now consider A 6= O. We need to show that kAk > 0. There must exist a column of A that is not 0. We call this column A∗k and take x = ek . Then kek k=1 kAk = max kAxk ≥ kAek k kxk=1
[email protected] MATH 532 25 Matrix Norms
Proof 1 kAk ≥ 0 is obvious since this holds for the vector norm. It remains to show that kAk = 0 if and only if A = O. Assume A = O, then
kAk = max k Ax k = 0. kxk=1 |{z} =0
So now consider A 6= O. We need to show that kAk > 0. There must exist a column of A that is not 0. We call this column A∗k and take x = ek . Then kek k=1 kAk = max kAxk ≥ kAek k = kA∗k k > 0 kxk=1
since A∗k 6= 0.
[email protected] MATH 532 25 = max k(A + B)xk = max kAx + Bxk ≤ max (kAxk + kBxk) = max kAxk + max kBxk = kAk + kBk.
= max kαAxk = |α| max kAxk = |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk
[email protected] MATH 532 26 = max k(A + B)xk = max kAx + Bxk ≤ max (kAxk + kBxk) = max kAxk + max kBxk = kAk + kBk.
= |α| max kAxk = |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk
[email protected] MATH 532 26 = max k(A + B)xk = max kAx + Bxk ≤ max (kAxk + kBxk) = max kAxk + max kBxk = kAk + kBk.
= |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk = |α| max kAxk
[email protected] MATH 532 26 = max k(A + B)xk = max kAx + Bxk ≤ max (kAxk + kBxk) = max kAxk + max kBxk = kAk + kBk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk = |α| max kAxk = |α|kAk.
[email protected] MATH 532 26 = max k(A + B)xk = max kAx + Bxk ≤ max (kAxk + kBxk) = max kAxk + max kBxk = kAk + kBk.
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk = |α| max kAxk = |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk
[email protected] MATH 532 26 = max kAx + Bxk ≤ max (kAxk + kBxk) = max kAxk + max kBxk = kAk + kBk.
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk = |α| max kAxk = |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk = max k(A + B)xk
[email protected] MATH 532 26 ≤ max (kAxk + kBxk) = max kAxk + max kBxk = kAk + kBk.
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk = |α| max kAxk = |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk = max k(A + B)xk = max kAx + Bxk
[email protected] MATH 532 26 = max kAxk + max kBxk = kAk + kBk.
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk = |α| max kAxk = |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk = max k(A + B)xk = max kAx + Bxk ≤ max (kAxk + kBxk)
[email protected] MATH 532 26 = kAk + kBk.
Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk = |α| max kAxk = |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk = max k(A + B)xk = max kAx + Bxk ≤ max (kAxk + kBxk) = max kAxk + max kBxk
[email protected] MATH 532 26 Matrix Norms
Proof (cont.) 2 Using the corresponding property for the vector norm we have
kαAk = max kαAxk = |α| max kAxk = |α|kAk.
3 Also straightforward (based on the triangle inequality for the vector norm)
kA + Bk = max k(A + B)xk = max kAx + Bxk ≤ max (kAxk + kBxk) = max kAxk + max kBxk = kAk + kBk.
[email protected] MATH 532 26 = kAByk (for some y with kyk = 1)
(1) (1) ≤ kAkkByk ≤ kAkkBk kyk . |{z} =1
and so kAxk kAxk kAk = max kAxk = max ≥ . kxk=1 x6=0 kxk kxk Therefore kAxk ≤ kAkkxk. (1) But then we also have kABk ≤ kAkkBk since
kABk = max kABxk kxk=1
Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk
[email protected] MATH 532 27 = kAByk (for some y with kyk = 1)
(1) (1) ≤ kAkkByk ≤ kAkkBk kyk . |{z} =1
kAxk kAxk = max kAxk = max ≥ . kxk=1 x6=0 kxk kxk Therefore kAxk ≤ kAkkxk. (1) But then we also have kABk ≤ kAkkBk since
kABk = max kABxk kxk=1
Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk and so kAk
[email protected] MATH 532 27 = kAByk (for some y with kyk = 1)
(1) (1) ≤ kAkkByk ≤ kAkkBk kyk . |{z} =1
kAxk ≥ . kxk Therefore kAxk ≤ kAkkxk. (1) But then we also have kABk ≤ kAkkBk since
kABk = max kABxk kxk=1
Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk and so kAxk kAk = max kAxk = max kxk=1 x6=0 kxk
[email protected] MATH 532 27 = kAByk (for some y with kyk = 1)
(1) (1) ≤ kAkkByk ≤ kAkkBk kyk . |{z} =1
Therefore kAxk ≤ kAkkxk. (1) But then we also have kABk ≤ kAkkBk since
kABk = max kABxk kxk=1
Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk and so kAxk kAxk kAk = max kAxk = max ≥ . kxk=1 x6=0 kxk kxk
[email protected] MATH 532 27 = kAByk (for some y with kyk = 1)
(1) (1) ≤ kAkkByk ≤ kAkkBk kyk . |{z} =1
But then we also have kABk ≤ kAkkBk since
kABk = max kABxk kxk=1
Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk and so kAxk kAxk kAk = max kAxk = max ≥ . kxk=1 x6=0 kxk kxk Therefore kAxk ≤ kAkkxk. (1)
[email protected] MATH 532 27 = kAByk (for some y with kyk = 1)
(1) (1) ≤ kAkkByk ≤ kAkkBk kyk . |{z} =1
kABk = max kABxk kxk=1
Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk and so kAxk kAxk kAk = max kAxk = max ≥ . kxk=1 x6=0 kxk kxk Therefore kAxk ≤ kAkkxk. (1) But then we also have kABk ≤ kAkkBk since
[email protected] MATH 532 27 (1) (1) ≤ kAkkByk ≤ kAkkBk kyk . |{z} =1
Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk and so kAxk kAxk kAk = max kAxk = max ≥ . kxk=1 x6=0 kxk kxk Therefore kAxk ≤ kAkkxk. (1) But then we also have kABk ≤ kAkkBk since
kABk = max kABxk = kAByk (for some y with kyk = 1) kxk=1
[email protected] MATH 532 27 (1) ≤ kAkkBk kyk . |{z} =1
Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk and so kAxk kAxk kAk = max kAxk = max ≥ . kxk=1 x6=0 kxk kxk Therefore kAxk ≤ kAkkxk. (1) But then we also have kABk ≤ kAkkBk since
kABk = max kABxk = kAByk (for some y with kyk = 1) kxk=1 (1) ≤ kAkkByk
[email protected] MATH 532 27 Matrix Norms Proof (cont.) 4 First note that kAxk max kAxk = max kxk=1 x6=0 kxk and so kAxk kAxk kAk = max kAxk = max ≥ . kxk=1 x6=0 kxk kxk Therefore kAxk ≤ kAkkxk. (1) But then we also have kABk ≤ kAkkBk since
kABk = max kABxk = kAByk (for some y with kyk = 1) kxk=1 (1) (1) ≤ kAkkByk ≤ kAkkBk kyk . |{z} =1
[email protected] MATH 532 27 The induced matrix norm can be interpreted geometrically: kAk: the most a vector on the unit sphere can be stretched when transformed by A. 1 kA−1k : the most a vector on the unit sphere can be shrunk when transformed by A.
Figure: induced matrix 2-norm from [Mey00].
Matrix Norms Remark One can show (see HW) that — if A is invertible —
1 min kAxk = . kxk=1 kA−1k
[email protected] MATH 532 28 Matrix Norms Remark One can show (see HW) that — if A is invertible —
1 min kAxk = . kxk=1 kA−1k
The induced matrix norm can be interpreted geometrically: kAk: the most a vector on the unit sphere can be stretched when transformed by A. 1 kA−1k : the most a vector on the unit sphere can be shrunk when transformed by A.
[email protected] MATH 532 28
Figure: induced matrix 2-norm from [Mey00]. Remark We also have p λmax = σ1, the largest singular value of A, p λmin = σn, the smallest singular value of A.
Matrix Norms Matrix 2-norm
Theorem Let A be an m × n matrix. Then p 1 kAk2 = max kAxk2 = λmax. kxk=1
−1 1 1 2 kA k2 = = √ . minkxk=1 kAxk2 λmin T where λmax and λmin are the largest and smallest eigenvalues of A A, respectively.
[email protected] MATH 532 29 Matrix Norms Matrix 2-norm
Theorem Let A be an m × n matrix. Then p 1 kAk2 = max kAxk2 = λmax. kxk=1
−1 1 1 2 kA k2 = = √ . minkxk=1 kAxk2 λmin T where λmax and λmin are the largest and smallest eigenvalues of A A, respectively.
Remark We also have p λmax = σ1, the largest singular value of A, p λmin = σn, the smallest singular value of A.
[email protected] MATH 532 29 The idea is to solve a constrained optimization problem (as in calculus), i.e.,
2 T maximize f (x) = kAxk2 = (Ax) Ax 2 T subject to g(x) = kxk2 = x x = 1.
We do this by introducing a Lagrange multiplier λ and define
h(x, λ) = f (x) − λg(x) = xT AT Ax − λxT x.
Matrix Norms
Proof We will show only (1), the largest singular value ((2) goes similarly).
[email protected] MATH 532 30 We do this by introducing a Lagrange multiplier λ and define
h(x, λ) = f (x) − λg(x) = xT AT Ax − λxT x.
Matrix Norms
Proof We will show only (1), the largest singular value ((2) goes similarly).
The idea is to solve a constrained optimization problem (as in calculus), i.e.,
2 T maximize f (x) = kAxk2 = (Ax) Ax 2 T subject to g(x) = kxk2 = x x = 1.
[email protected] MATH 532 30 Matrix Norms
Proof We will show only (1), the largest singular value ((2) goes similarly).
The idea is to solve a constrained optimization problem (as in calculus), i.e.,
2 T maximize f (x) = kAxk2 = (Ax) Ax 2 T subject to g(x) = kxk2 = x x = 1.
We do this by introducing a Lagrange multiplier λ and define
h(x, λ) = f (x) − λg(x) = xT AT Ax − λxT x.
[email protected] MATH 532 30 ∂xT ∂x ∂xT ∂x = AT Ax + xT AT A − λ x − λxT ∂xi ∂xi ∂xi ∂xi T T T = 2ei A Ax − 2λei x T = 2 (A Ax)i − (λx)i , i = 1,..., n.
Together this yields AT Ax − λx = 0 ⇐⇒ AT A − λI x = 0,
so that λ must be an eigenvalue of AT A (since g(x) = xT x = 1 ensures x 6= 0).
∂ xT AT Ax − λxT x ∂xi
Matrix Norms
Proof (cont.) Necessary and sufficient (since quadratic) condition for maximum: ∂h = 0, i = 1,..., n, g(x) = 1 ∂xi
[email protected] MATH 532 31 ∂xT ∂x ∂xT ∂x = AT Ax + xT AT A − λ x − λxT ∂xi ∂xi ∂xi ∂xi T T T = 2ei A Ax − 2λei x T = 2 (A Ax)i − (λx)i , i = 1,..., n.
Together this yields AT Ax − λx = 0 ⇐⇒ AT A − λI x = 0,
so that λ must be an eigenvalue of AT A (since g(x) = xT x = 1 ensures x 6= 0).
Matrix Norms
Proof (cont.) Necessary and sufficient (since quadratic) condition for maximum: ∂h = 0, i = 1,..., n, g(x) = 1 ∂xi
∂ xT AT Ax − λxT x ∂xi
[email protected] MATH 532 31 T T T = 2ei A Ax − 2λei x T = 2 (A Ax)i − (λx)i , i = 1,..., n.
Together this yields AT Ax − λx = 0 ⇐⇒ AT A − λI x = 0,
so that λ must be an eigenvalue of AT A (since g(x) = xT x = 1 ensures x 6= 0).
Matrix Norms
Proof (cont.) Necessary and sufficient (since quadratic) condition for maximum: ∂h = 0, i = 1,..., n, g(x) = 1 ∂xi
∂ ∂xT ∂x ∂xT ∂x xT AT Ax − λxT x = AT Ax + xT AT A − λ x − λxT ∂xi ∂xi ∂xi ∂xi ∂xi
[email protected] MATH 532 31 T = 2 (A Ax)i − (λx)i , i = 1,..., n.
Together this yields AT Ax − λx = 0 ⇐⇒ AT A − λI x = 0,
so that λ must be an eigenvalue of AT A (since g(x) = xT x = 1 ensures x 6= 0).
Matrix Norms
Proof (cont.) Necessary and sufficient (since quadratic) condition for maximum: ∂h = 0, i = 1,..., n, g(x) = 1 ∂xi
∂ ∂xT ∂x ∂xT ∂x xT AT Ax − λxT x = AT Ax + xT AT A − λ x − λxT ∂xi ∂xi ∂xi ∂xi ∂xi T T T = 2ei A Ax − 2λei x
[email protected] MATH 532 31 Together this yields AT Ax − λx = 0 ⇐⇒ AT A − λI x = 0,
so that λ must be an eigenvalue of AT A (since g(x) = xT x = 1 ensures x 6= 0).
Matrix Norms
Proof (cont.) Necessary and sufficient (since quadratic) condition for maximum: ∂h = 0, i = 1,..., n, g(x) = 1 ∂xi
∂ ∂xT ∂x ∂xT ∂x xT AT Ax − λxT x = AT Ax + xT AT A − λ x − λxT ∂xi ∂xi ∂xi ∂xi ∂xi T T T = 2ei A Ax − 2λei x T = 2 (A Ax)i − (λx)i , i = 1,..., n.
[email protected] MATH 532 31 so that λ must be an eigenvalue of AT A (since g(x) = xT x = 1 ensures x 6= 0).
Matrix Norms
Proof (cont.) Necessary and sufficient (since quadratic) condition for maximum: ∂h = 0, i = 1,..., n, g(x) = 1 ∂xi
∂ ∂xT ∂x ∂xT ∂x xT AT Ax − λxT x = AT Ax + xT AT A − λ x − λxT ∂xi ∂xi ∂xi ∂xi ∂xi T T T = 2ei A Ax − 2λei x T = 2 (A Ax)i − (λx)i , i = 1,..., n.
Together this yields AT Ax − λx = 0 ⇐⇒ AT A − λI x = 0,
[email protected] MATH 532 31 Matrix Norms
Proof (cont.) Necessary and sufficient (since quadratic) condition for maximum: ∂h = 0, i = 1,..., n, g(x) = 1 ∂xi
∂ ∂xT ∂x ∂xT ∂x xT AT Ax − λxT x = AT Ax + xT AT A − λ x − λxT ∂xi ∂xi ∂xi ∂xi ∂xi T T T = 2ei A Ax − 2λei x T = 2 (A Ax)i − (λx)i , i = 1,..., n.
Together this yields AT Ax − λx = 0 ⇐⇒ AT A − λI x = 0,
so that λ must be an eigenvalue of AT A (since g(x) = xT x = 1 ensures x 6= 0).
[email protected] MATH 532 31 = max kAxk2 2 kxk2=1 √ p = max λ = λmax.
First, AT Ax = λx =⇒ xT AT Ax = λxT x = λ
so that √ √ T T kAxk2 = x A Ax = λ. And then
kAk2 = max kAxk2 kxk2=1
Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue.
[email protected] MATH 532 32 = max kAxk2 2 kxk2=1 √ p = max λ = λmax.
=⇒ xT AT Ax = λxT x = λ
so that √ √ T T kAxk2 = x A Ax = λ. And then
kAk2 = max kAxk2 kxk2=1
Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue. First, AT Ax = λx
[email protected] MATH 532 32 = max kAxk2 2 kxk2=1 √ p = max λ = λmax.
= λxT x = λ
so that √ √ T T kAxk2 = x A Ax = λ. And then
kAk2 = max kAxk2 kxk2=1
Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue. First, AT Ax = λx =⇒ xT AT Ax
[email protected] MATH 532 32 = max kAxk2 2 kxk2=1 √ p = max λ = λmax.
= λ
so that √ √ T T kAxk2 = x A Ax = λ. And then
kAk2 = max kAxk2 kxk2=1
Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue. First, AT Ax = λx =⇒ xT AT Ax = λxT x
[email protected] MATH 532 32 = max kAxk2 2 kxk2=1 √ p = max λ = λmax.
so that √ √ T T kAxk2 = x A Ax = λ. And then
kAk2 = max kAxk2 kxk2=1
Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue. First, AT Ax = λx =⇒ xT AT Ax = λxT x = λ
[email protected] MATH 532 32 = max kAxk2 2 kxk2=1 √ p = max λ = λmax.
And then
kAk2 = max kAxk2 kxk2=1
Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue. First, AT Ax = λx =⇒ xT AT Ax = λxT x = λ so that √ √ T T kAxk2 = x A Ax = λ.
[email protected] MATH 532 32 = max kAxk2 2 kxk2=1 √ p = max λ = λmax.
Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue. First, AT Ax = λx =⇒ xT AT Ax = λxT x = λ so that √ √ T T kAxk2 = x A Ax = λ. And then
kAk2 = max kAxk2 kxk2=1
[email protected] MATH 532 32 √ p = max λ = λmax.
Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue. First, AT Ax = λx =⇒ xT AT Ax = λxT x = λ so that √ √ T T kAxk2 = x A Ax = λ. And then
kAk2 = max kAxk2 = max kAxk2 kxk =1 2 2 kxk2=1
[email protected] MATH 532 32 Matrix Norms
Proof (cont.) In fact, as we now show, λ is the maximal eigenvalue. First, AT Ax = λx =⇒ xT AT Ax = λxT x = λ so that √ √ T T kAxk2 = x A Ax = λ. And then
kAk2 = max kAxk2 = max kAxk2 kxk =1 2 2 kxk2=1 √ p = max λ = λmax.
[email protected] MATH 532 32 T 2 kAk2 = kA k2
3 T 2 T kA Ak2 = kAk2 = kAA k2 AO 4 = max {kAk2, kBk2} OB T T T 5 kU AVk2 = kAk2 provided UU = I and V V = I (orthogonal matrices).
Remark The proof is a HW problem.
Matrix Norms Special properties of the 2-norm
T 1 kAk2 = max max |y Ax| kxk2=1 kyk2=1
[email protected] MATH 532 33 3 T 2 T kA Ak2 = kAk2 = kAA k2 AO 4 = max {kAk2, kBk2} OB T T T 5 kU AVk2 = kAk2 provided UU = I and V V = I (orthogonal matrices).
Remark The proof is a HW problem.
Matrix Norms Special properties of the 2-norm
T 1 kAk2 = max max |y Ax| kxk2=1 kyk2=1 T 2 kAk2 = kA k2
[email protected] MATH 532 33 AO 4 = max {kAk2, kBk2} OB T T T 5 kU AVk2 = kAk2 provided UU = I and V V = I (orthogonal matrices).
Remark The proof is a HW problem.
Matrix Norms Special properties of the 2-norm
T 1 kAk2 = max max |y Ax| kxk2=1 kyk2=1 T 2 kAk2 = kA k2
3 T 2 T kA Ak2 = kAk2 = kAA k2
[email protected] MATH 532 33 T T T 5 kU AVk2 = kAk2 provided UU = I and V V = I (orthogonal matrices).
Remark The proof is a HW problem.
Matrix Norms Special properties of the 2-norm
T 1 kAk2 = max max |y Ax| kxk2=1 kyk2=1 T 2 kAk2 = kA k2
3 T 2 T kA Ak2 = kAk2 = kAA k2 AO 4 = max {kAk2, kBk2} OB
[email protected] MATH 532 33 Remark The proof is a HW problem.
Matrix Norms Special properties of the 2-norm
T 1 kAk2 = max max |y Ax| kxk2=1 kyk2=1 T 2 kAk2 = kA k2
3 T 2 T kA Ak2 = kAk2 = kAA k2 AO 4 = max {kAk2, kBk2} OB T T T 5 kU AVk2 = kAk2 provided UU = I and V V = I (orthogonal matrices).
[email protected] MATH 532 33 Matrix Norms Special properties of the 2-norm
T 1 kAk2 = max max |y Ax| kxk2=1 kyk2=1 T 2 kAk2 = kA k2
3 T 2 T kA Ak2 = kAk2 = kAA k2 AO 4 = max {kAk2, kBk2} OB T T T 5 kU AVk2 = kAk2 provided UU = I and V V = I (orthogonal matrices).
Remark The proof is a HW problem.
[email protected] MATH 532 33 Remark We know these are norms, so what we need to do is verify that the formulas hold. We will show (1).
Matrix Norms Matrix 1-norm and ∞-norm
Theorem Let A be an m × n matrix. Then we have 1 the column sum norm
m X kAk1 = max kAxk1 = max |aij |, kxk =1 j=1,...,n 1 i=1
2 and the row sum norm
n X kAk∞ = max kAxk∞ = max |aij |. kxk∞=1 i=1,...,m j=1
[email protected] MATH 532 34 Matrix Norms Matrix 1-norm and ∞-norm
Theorem Let A be an m × n matrix. Then we have 1 the column sum norm
m X kAk1 = max kAxk1 = max |aij |, kxk =1 j=1,...,n 1 i=1
2 and the row sum norm
n X kAk∞ = max kAxk∞ = max |aij |. kxk∞=1 i=1,...,m j=1
Remark We know these are norms, so what we need to do is verify that the formulas hold. We will show (1).
[email protected] MATH 532 34 m m m n X X X X = |(Ax)i | = |Ai∗x| = | aij xj | i=1 i=1 i=1 j=1 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1 n " m # " m # n X X X X = |xj | |aij | ≤ max |aij | |xj |. j=1,...,n j=1 i=1 i=1 j=1
Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
m X kAxk1 ≤ max |aij |. j=1,...,n i=1
kAxk1
Matrix Norms Proof
First we look at kAxk1.
[email protected] MATH 532 35 m m n X X X = |Ai∗x| = | aij xj | i=1 i=1 j=1 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1 n " m # " m # n X X X X = |xj | |aij | ≤ max |aij | |xj |. j=1,...,n j=1 i=1 i=1 j=1
Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
m X kAxk1 ≤ max |aij |. j=1,...,n i=1
Matrix Norms Proof
First we look at kAxk1.
m X kAxk1 = |(Ax)i | i=1
[email protected] MATH 532 35 m n X X = | aij xj | i=1 j=1 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1 n " m # " m # n X X X X = |xj | |aij | ≤ max |aij | |xj |. j=1,...,n j=1 i=1 i=1 j=1
Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
m X kAxk1 ≤ max |aij |. j=1,...,n i=1
Matrix Norms Proof
First we look at kAxk1.
m m X X kAxk1 = |(Ax)i | = |Ai∗x| i=1 i=1
[email protected] MATH 532 35 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1 n " m # " m # n X X X X = |xj | |aij | ≤ max |aij | |xj |. j=1,...,n j=1 i=1 i=1 j=1
Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
m X kAxk1 ≤ max |aij |. j=1,...,n i=1
Matrix Norms Proof
First we look at kAxk1.
m m m n X X X X kAxk1 = |(Ax)i | = |Ai∗x| = | aij xj | i=1 i=1 i=1 j=1
[email protected] MATH 532 35 n " m # " m # n X X X X = |xj | |aij | ≤ max |aij | |xj |. j=1,...,n j=1 i=1 i=1 j=1
Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
m X kAxk1 ≤ max |aij |. j=1,...,n i=1
Matrix Norms Proof
First we look at kAxk1.
m m m n X X X X kAxk1 = |(Ax)i | = |Ai∗x| = | aij xj | i=1 i=1 i=1 j=1 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1
[email protected] MATH 532 35 " m # n X X ≤ max |aij | |xj |. j=1,...,n i=1 j=1
Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
m X kAxk1 ≤ max |aij |. j=1,...,n i=1
Matrix Norms Proof
First we look at kAxk1.
m m m n X X X X kAxk1 = |(Ax)i | = |Ai∗x| = | aij xj | i=1 i=1 i=1 j=1 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1 n " m # X X = |xj | |aij | j=1 i=1
[email protected] MATH 532 35 Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
m X kAxk1 ≤ max |aij |. j=1,...,n i=1
Matrix Norms Proof
First we look at kAxk1.
m m m n X X X X kAxk1 = |(Ax)i | = |Ai∗x| = | aij xj | i=1 i=1 i=1 j=1 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1 n " m # " m # n X X X X = |xj | |aij | ≤ max |aij | |xj |. j=1,...,n j=1 i=1 i=1 j=1
[email protected] MATH 532 35 m X kAxk1 ≤ max |aij |. j=1,...,n i=1
Matrix Norms Proof
First we look at kAxk1.
m m m n X X X X kAxk1 = |(Ax)i | = |Ai∗x| = | aij xj | i=1 i=1 i=1 j=1 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1 n " m # " m # n X X X X = |xj | |aij | ≤ max |aij | |xj |. j=1,...,n j=1 i=1 i=1 j=1
Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
[email protected] MATH 532 35 Matrix Norms Proof
First we look at kAxk1.
m m m n X X X X kAxk1 = |(Ax)i | = |Ai∗x| = | aij xj | i=1 i=1 i=1 j=1 m n reg.∆ X X ≤ |aij | |xj | i=1 j=1 n " m # " m # n X X X X = |xj | |aij | ≤ max |aij | |xj |. j=1,...,n j=1 i=1 i=1 j=1
Since we actually need to look at kAxk1 for kxk1 = 1 we note that Pn kxk1 = j=1 |xj | and therefore have
m X kAxk1 ≤ max |aij |. j=1,...,n i=1
[email protected] MATH 532 35 m X = kA∗k k1 = |aik | i=1 m X = max |aik | j=1...,n i=1 due to our choice of k. Since kek k1 = 1 we indeed have the desired formula.
kAxk1 = kAek k1
Matrix Norms
Proof (cont.)
We even have equality since for x = ek , where k is the index such that A∗k has maximum column sum, we get
[email protected] MATH 532 36 m X = kA∗k k1 = |aik | i=1 m X = max |aik | j=1...,n i=1 due to our choice of k. Since kek k1 = 1 we indeed have the desired formula.
Matrix Norms
Proof (cont.)
We even have equality since for x = ek , where k is the index such that A∗k has maximum column sum, we get
kAxk1 = kAek k1
[email protected] MATH 532 36 m X = |aik | i=1 m X = max |aik | j=1...,n i=1 due to our choice of k. Since kek k1 = 1 we indeed have the desired formula.
Matrix Norms
Proof (cont.)
We even have equality since for x = ek , where k is the index such that A∗k has maximum column sum, we get
kAxk1 = kAek k1 = kA∗k k1
[email protected] MATH 532 36 m X = max |aik | j=1...,n i=1 due to our choice of k. Since kek k1 = 1 we indeed have the desired formula.
Matrix Norms
Proof (cont.)
We even have equality since for x = ek , where k is the index such that A∗k has maximum column sum, we get
m X kAxk1 = kAek k1 = kA∗k k1 = |aik | i=1
[email protected] MATH 532 36 Since kek k1 = 1 we indeed have the desired formula.
Matrix Norms
Proof (cont.)
We even have equality since for x = ek , where k is the index such that A∗k has maximum column sum, we get
m X kAxk1 = kAek k1 = kA∗k k1 = |aik | i=1 m X = max |aik | j=1...,n i=1 due to our choice of k.
[email protected] MATH 532 36 Matrix Norms
Proof (cont.)
We even have equality since for x = ek , where k is the index such that A∗k has maximum column sum, we get
m X kAxk1 = kAek k1 = kA∗k k1 = |aik | i=1 m X = max |aik | j=1...,n i=1 due to our choice of k. Since kek k1 = 1 we indeed have the desired formula.
[email protected] MATH 532 36 Inner Product Spaces Outline
1 Vector Norms
2 Matrix Norms
3 Inner Product Spaces
4 Orthogonal Vectors
5 Gram–Schmidt Orthogonalization & QR Factorization
6 Unitary and Orthogonal Matrices
7 Orthogonal Reduction
8 Complementary Subspaces
9 Orthogonal Decomposition
10 Singular Value Decomposition
11 Orthogonal Projections
[email protected] MATH 532 37 Remark The following two properties (providing bilinearity) are implied (see HW)
hαx, yi = αhx, yi hx + y, zi = hx, zi + hy, zi.
Inner Product Spaces
Definition A general inner product in a real (complex) vector space V is a symmetric (Hermitian) bilinear form h·, ·i : V × V → R (C), i.e., 1 hx, xi ∈ R≥0 with hx, xi = 0 if and only if x = 0. 2 hx, αyi = αhx, yi for all scalars α. 3 hx, y + zi = hx, yi + hx, zi. 4 hx, yi = hy, xi (or hx, yi = hy, xi if complex).
[email protected] MATH 532 38 Inner Product Spaces
Definition A general inner product in a real (complex) vector space V is a symmetric (Hermitian) bilinear form h·, ·i : V × V → R (C), i.e., 1 hx, xi ∈ R≥0 with hx, xi = 0 if and only if x = 0. 2 hx, αyi = αhx, yi for all scalars α. 3 hx, y + zi = hx, yi + hx, zi. 4 hx, yi = hy, xi (or hx, yi = hy, xi if complex).
Remark The following two properties (providing bilinearity) are implied (see HW)
hαx, yi = αhx, yi hx + y, zi = hx, zi + hy, zi.
[email protected] MATH 532 38 One can show (analogous to the Euclidean case) that k · k is a norm.
In particular, we have a general Cauchy–Schwarz–Bunyakovsky inequality |hx, yi| ≤ kxk kyk.
Inner Product Spaces
As before, any inner product induces a norm via
k · k = ph·, ·i.
[email protected] MATH 532 39 In particular, we have a general Cauchy–Schwarz–Bunyakovsky inequality |hx, yi| ≤ kxk kyk.
Inner Product Spaces
As before, any inner product induces a norm via
k · k = ph·, ·i.
One can show (analogous to the Euclidean case) that k · k is a norm.
[email protected] MATH 532 39 Inner Product Spaces
As before, any inner product induces a norm via
k · k = ph·, ·i.
One can show (analogous to the Euclidean case) that k · k is a norm.
In particular, we have a general Cauchy–Schwarz–Bunyakovsky inequality |hx, yi| ≤ kxk kyk.
[email protected] MATH 532 39 2 For nonsingular matrices A we get the A-inner product on Rn, i.e.,
hx, yi = xT AT Ay
with √ p T T kxkA = hx, xi = x A Ax = kAxk2.
× × 3 If V = Rm n (or Cm n) then we get the standard inner product for matrices, i.e.,
hA, Bi = trace(AT B) (or trace(A∗B))
with q p T kAk = hA, Ai = trace(A A) = kAkF .
Inner Product Spaces Example ∗ 1 hx, yi = xT y (or x y), the standard inner product for Rn (Cn).
[email protected] MATH 532 40 with √ p T T kxkA = hx, xi = x A Ax = kAxk2.
× × 3 If V = Rm n (or Cm n) then we get the standard inner product for matrices, i.e.,
hA, Bi = trace(AT B) (or trace(A∗B))
with q p T kAk = hA, Ai = trace(A A) = kAkF .
Inner Product Spaces Example ∗ 1 hx, yi = xT y (or x y), the standard inner product for Rn (Cn). 2 For nonsingular matrices A we get the A-inner product on Rn, i.e.,
hx, yi = xT AT Ay
[email protected] MATH 532 40 × × 3 If V = Rm n (or Cm n) then we get the standard inner product for matrices, i.e.,
hA, Bi = trace(AT B) (or trace(A∗B))
with q p T kAk = hA, Ai = trace(A A) = kAkF .
Inner Product Spaces Example ∗ 1 hx, yi = xT y (or x y), the standard inner product for Rn (Cn). 2 For nonsingular matrices A we get the A-inner product on Rn, i.e.,
hx, yi = xT AT Ay
with √ p T T kxkA = hx, xi = x A Ax = kAxk2.
[email protected] MATH 532 40 with q p T kAk = hA, Ai = trace(A A) = kAkF .
Inner Product Spaces Example ∗ 1 hx, yi = xT y (or x y), the standard inner product for Rn (Cn). 2 For nonsingular matrices A we get the A-inner product on Rn, i.e.,
hx, yi = xT AT Ay
with √ p T T kxkA = hx, xi = x A Ax = kAxk2.
× × 3 If V = Rm n (or Cm n) then we get the standard inner product for matrices, i.e.,
hA, Bi = trace(AT B) (or trace(A∗B))
[email protected] MATH 532 40 = kAkF .
Inner Product Spaces Example ∗ 1 hx, yi = xT y (or x y), the standard inner product for Rn (Cn). 2 For nonsingular matrices A we get the A-inner product on Rn, i.e.,
hx, yi = xT AT Ay
with √ p T T kxkA = hx, xi = x A Ax = kAxk2.
× × 3 If V = Rm n (or Cm n) then we get the standard inner product for matrices, i.e.,
hA, Bi = trace(AT B) (or trace(A∗B))
with p q kAk = hA, Ai = trace(AT A)
[email protected] MATH 532 40 Inner Product Spaces Example ∗ 1 hx, yi = xT y (or x y), the standard inner product for Rn (Cn). 2 For nonsingular matrices A we get the A-inner product on Rn, i.e.,
hx, yi = xT AT Ay
with √ p T T kxkA = hx, xi = x A Ax = kAxk2.
× × 3 If V = Rm n (or Cm n) then we get the standard inner product for matrices, i.e.,
hA, Bi = trace(AT B) (or trace(A∗B))
with q p T kAk = hA, Ai = trace(A A) = kAkF .
[email protected] MATH 532 40 Z b hf , gi = f (t)g(t)dt a with 1/2 Z b ! p 2 kf k = hf , f i = (f (t)) dt . a
Inner Product Spaces
Remark In the infinite-dimensional setting we have, e.g., for f , g continuous functions on (a, b)
[email protected] MATH 532 41 with 1/2 Z b ! p 2 kf k = hf , f i = (f (t)) dt . a
Inner Product Spaces
Remark In the infinite-dimensional setting we have, e.g., for f , g continuous functions on (a, b) Z b hf , gi = f (t)g(t)dt a
[email protected] MATH 532 41 Inner Product Spaces
Remark In the infinite-dimensional setting we have, e.g., for f , g continuous functions on (a, b) Z b hf , gi = f (t)g(t)dt a with 1/2 Z b ! p 2 kf k = hf , f i = (f (t)) dt . a
[email protected] MATH 532 41 = hx + y, x + yi + hx − y, x − yi = hx, xi + hx, yi + hy, xi + hy, yi + hx, xi − hx, yi − hy, xi + hy, yi = 2hx, xi + 2hy, yi = 2 kxk2 + kyk2 .
This is true since
kx + yk2 + kx − yk2
Inner Product Spaces Parallelogram identity
In any inner product space the so-called parallelogram identity holds, i.e., kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 . (2)
[email protected] MATH 532 42 = hx + y, x + yi + hx − y, x − yi = hx, xi + hx, yi + hy, xi + hy, yi + hx, xi − hx, yi − hy, xi + hy, yi = 2hx, xi + 2hy, yi = 2 kxk2 + kyk2 .
Inner Product Spaces Parallelogram identity
In any inner product space the so-called parallelogram identity holds, i.e., kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 . (2)
This is true since
kx + yk2 + kx − yk2
[email protected] MATH 532 42 = hx, xi + hx, yi + hy, xi + hy, yi + hx, xi − hx, yi − hy, xi + hy, yi = 2hx, xi + 2hy, yi = 2 kxk2 + kyk2 .
Inner Product Spaces Parallelogram identity
In any inner product space the so-called parallelogram identity holds, i.e., kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 . (2)
This is true since
kx + yk2 + kx − yk2 = hx + y, x + yi + hx − y, x − yi
[email protected] MATH 532 42 = 2hx, xi + 2hy, yi = 2 kxk2 + kyk2 .
Inner Product Spaces Parallelogram identity
In any inner product space the so-called parallelogram identity holds, i.e., kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 . (2)
This is true since
kx + yk2 + kx − yk2 = hx + y, x + yi + hx − y, x − yi = hx, xi + hx, yi + hy, xi + hy, yi + hx, xi − hx, yi − hy, xi + hy, yi
[email protected] MATH 532 42 = 2 kxk2 + kyk2 .
Inner Product Spaces Parallelogram identity
In any inner product space the so-called parallelogram identity holds, i.e., kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 . (2)
This is true since
kx + yk2 + kx − yk2 = hx + y, x + yi + hx − y, x − yi = hx, xi + hx, yi + hy, xi + hy, yi + hx, xi − hx, yi − hy, xi + hy, yi = 2hx, xi + 2hy, yi
[email protected] MATH 532 42 Inner Product Spaces Parallelogram identity
In any inner product space the so-called parallelogram identity holds, i.e., kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 . (2)
This is true since
kx + yk2 + kx − yk2 = hx + y, x + yi + hx − y, x − yi = hx, xi + hx, yi + hy, xi + hy, yi + hx, xi − hx, yi − hy, xi + hy, yi = 2hx, xi + 2hy, yi = 2 kxk2 + kyk2 .
[email protected] MATH 532 42 but — if the parallelogram identity holds — then we can get an inner product from a norm (i.e., a Banach space becomes a Hilbert space).
Theorem Let V be a real vector space with norm k · k. If the parallelogram identity (2) holds then
1 hx, yi = kx + yk2 − kx − yk2 (3) 4 is an inner product on V.
Inner Product Spaces Polarization identity
The following theorem shows that we not only get a norm from an inner product (i.e., every Hilbert space is a Banach space),
[email protected] MATH 532 43 Theorem Let V be a real vector space with norm k · k. If the parallelogram identity (2) holds then
1 hx, yi = kx + yk2 − kx − yk2 (3) 4 is an inner product on V.
Inner Product Spaces Polarization identity
The following theorem shows that we not only get a norm from an inner product (i.e., every Hilbert space is a Banach space), but — if the parallelogram identity holds — then we can get an inner product from a norm (i.e., a Banach space becomes a Hilbert space).
[email protected] MATH 532 43 Inner Product Spaces Polarization identity
The following theorem shows that we not only get a norm from an inner product (i.e., every Hilbert space is a Banach space), but — if the parallelogram identity holds — then we can get an inner product from a norm (i.e., a Banach space becomes a Hilbert space).
Theorem Let V be a real vector space with norm k · k. If the parallelogram identity (2) holds then
1 hx, yi = kx + yk2 − kx − yk2 (3) 4 is an inner product on V.
[email protected] MATH 532 43 1 Nonnegativity:
1 1 hx, xi = kx + xk2 − kx − xk2 = k2xk2 = kxk2 ≥ 0. 4 4 Moreover, hx, xi > 0 if and only if x = 0 since hx, xi = kxk2. 4 Symmetry: hx, yi = hy, xi is clear since kx − yk = ky − xk.
Inner Product Spaces
Proof We need to show that all four properties of a general inner product hold.
[email protected] MATH 532 44 1 1 = kx + xk2 − kx − xk2 = k2xk2 = kxk2 ≥ 0. 4 4 Moreover, hx, xi > 0 if and only if x = 0 since hx, xi = kxk2. 4 Symmetry: hx, yi = hy, xi is clear since kx − yk = ky − xk.
Inner Product Spaces
Proof We need to show that all four properties of a general inner product hold. 1 Nonnegativity:
hx, xi
[email protected] MATH 532 44 1 = k2xk2 = kxk2 ≥ 0. 4 Moreover, hx, xi > 0 if and only if x = 0 since hx, xi = kxk2. 4 Symmetry: hx, yi = hy, xi is clear since kx − yk = ky − xk.
Inner Product Spaces
Proof We need to show that all four properties of a general inner product hold. 1 Nonnegativity:
1 hx, xi = kx + xk2 − kx − xk2 4
[email protected] MATH 532 44 = kxk2 ≥ 0.
Moreover, hx, xi > 0 if and only if x = 0 since hx, xi = kxk2. 4 Symmetry: hx, yi = hy, xi is clear since kx − yk = ky − xk.
Inner Product Spaces
Proof We need to show that all four properties of a general inner product hold. 1 Nonnegativity:
1 1 hx, xi = kx + xk2 − kx − xk2 = k2xk2 4 4
[email protected] MATH 532 44 Moreover, hx, xi > 0 if and only if x = 0 since hx, xi = kxk2. 4 Symmetry: hx, yi = hy, xi is clear since kx − yk = ky − xk.
Inner Product Spaces
Proof We need to show that all four properties of a general inner product hold. 1 Nonnegativity:
1 1 hx, xi = kx + xk2 − kx − xk2 = k2xk2 = kxk2 ≥ 0. 4 4
[email protected] MATH 532 44 4 Symmetry: hx, yi = hy, xi is clear since kx − yk = ky − xk.
Inner Product Spaces
Proof We need to show that all four properties of a general inner product hold. 1 Nonnegativity:
1 1 hx, xi = kx + xk2 − kx − xk2 = k2xk2 = kxk2 ≥ 0. 4 4 Moreover, hx, xi > 0 if and only if x = 0 since hx, xi = kxk2.
[email protected] MATH 532 44 Inner Product Spaces
Proof We need to show that all four properties of a general inner product hold. 1 Nonnegativity:
1 1 hx, xi = kx + xk2 − kx − xk2 = k2xk2 = kxk2 ≥ 0. 4 4 Moreover, hx, xi > 0 if and only if x = 0 since hx, xi = kxk2. 4 Symmetry: hx, yi = hy, xi is clear since kx − yk = ky − xk.
[email protected] MATH 532 44 and 1 kx − yk2 + kx − zk2 = kx − y + x − zk2 + kz − yk2 . (5) 2 Subtracting (5) from (4) we get
kx + yk2 − kx − yk2+kx + zk2 − kx − zk2 1 = k2x + y + zk2 − k2x − y − zk2 . 2 (6)
Inner Product Spaces
Proof (cont.) 3 Additivity: The parallelogram identity implies
1 kx + yk2 + kx + zk2 = kx + y + x + zk2 + ky − zk2 . (4) 2
[email protected] MATH 532 45 Subtracting (5) from (4) we get
kx + yk2 − kx − yk2+kx + zk2 − kx − zk2 1 = k2x + y + zk2 − k2x − y − zk2 . 2 (6)
Inner Product Spaces
Proof (cont.) 3 Additivity: The parallelogram identity implies
1 kx + yk2 + kx + zk2 = kx + y + x + zk2 + ky − zk2 . (4) 2 and 1 kx − yk2 + kx − zk2 = kx − y + x − zk2 + kz − yk2 . (5) 2
[email protected] MATH 532 45 Inner Product Spaces
Proof (cont.) 3 Additivity: The parallelogram identity implies
1 kx + yk2 + kx + zk2 = kx + y + x + zk2 + ky − zk2 . (4) 2 and 1 kx − yk2 + kx − zk2 = kx − y + x − zk2 + kz − yk2 . (5) 2 Subtracting (5) from (4) we get
kx + yk2 − kx − yk2+kx + zk2 − kx − zk2 1 = k2x + y + zk2 − k2x − y − zk2 . 2 (6)
[email protected] MATH 532 45 (6) 1 = k2x + y + zk2 − k2x − y − zk2 8 2 2! 1 y + z y + z = x + − x − 2 2 2 polarization y + z = 2hx, i. 2 Setting z = 0 in (7) yields y hx, yi = 2hx, i (8) 2 since hx, zi = 0.
Inner Product Spaces
Proof (cont.) The specific form of the polarized inner product implies
1 hx, yi + hx, zi = kx + yk2 − kx − yk2 + kx + zk2 − kx − zk2 4
(7)
[email protected] MATH 532 46 2 2! 1 y + z y + z = x + − x − 2 2 2 polarization y + z = 2hx, i. 2 Setting z = 0 in (7) yields y hx, yi = 2hx, i (8) 2 since hx, zi = 0.
Inner Product Spaces
Proof (cont.) The specific form of the polarized inner product implies
1 hx, yi + hx, zi = kx + yk2 − kx − yk2 + kx + zk2 − kx − zk2 4 (6) 1 = k2x + y + zk2 − k2x − y − zk2 8
(7)
[email protected] MATH 532 46 polarization y + z = 2hx, i. 2 Setting z = 0 in (7) yields y hx, yi = 2hx, i (8) 2 since hx, zi = 0.
Inner Product Spaces
Proof (cont.) The specific form of the polarized inner product implies
1 hx, yi + hx, zi = kx + yk2 − kx − yk2 + kx + zk2 − kx − zk2 4 (6) 1 = k2x + y + zk2 − k2x − y − zk2 8 2 2! 1 y + z y + z = x + − x − 2 2 2
(7)
[email protected] MATH 532 46 Setting z = 0 in (7) yields y hx, yi = 2hx, i (8) 2 since hx, zi = 0.
Inner Product Spaces
Proof (cont.) The specific form of the polarized inner product implies
1 hx, yi + hx, zi = kx + yk2 − kx − yk2 + kx + zk2 − kx − zk2 4 (6) 1 = k2x + y + zk2 − k2x − y − zk2 8 2 2! 1 y + z y + z = x + − x − 2 2 2 polarization y + z = 2hx, i. (7) 2
[email protected] MATH 532 46 Inner Product Spaces
Proof (cont.) The specific form of the polarized inner product implies
1 hx, yi + hx, zi = kx + yk2 − kx − yk2 + kx + zk2 − kx − zk2 4 (6) 1 = k2x + y + zk2 − k2x − y − zk2 8 2 2! 1 y + z y + z = x + − x − 2 2 2 polarization y + z = 2hx, i. (7) 2 Setting z = 0 in (7) yields y hx, yi = 2hx, i (8) 2 since hx, zi = 0.
[email protected] MATH 532 46 Since (8) is true for any y ∈ V we can, in particular, set y = y + z so that we have y + z hx, y + zi = 2hx, i. 2 This, however, is the right-hand side of (7) so that we end up with
hx, y + zi = hx, yi + hx, zi,
as desired.
Inner Product Spaces
Proof (cont.) To summarize, we have y + z hx, yi + hx, zi = 2hx, i. (7) 2 and y hx, yi = 2hx, i. (8) 2
[email protected] MATH 532 47 This, however, is the right-hand side of (7) so that we end up with
hx, y + zi = hx, yi + hx, zi,
as desired.
Inner Product Spaces
Proof (cont.) To summarize, we have y + z hx, yi + hx, zi = 2hx, i. (7) 2 and y hx, yi = 2hx, i. (8) 2 Since (8) is true for any y ∈ V we can, in particular, set y = y + z so that we have y + z hx, y + zi = 2hx, i. 2
[email protected] MATH 532 47 Inner Product Spaces
Proof (cont.) To summarize, we have y + z hx, yi + hx, zi = 2hx, i. (7) 2 and y hx, yi = 2hx, i. (8) 2 Since (8) is true for any y ∈ V we can, in particular, set y = y + z so that we have y + z hx, y + zi = 2hx, i. 2 This, however, is the right-hand side of (7) so that we end up with
hx, y + zi = hx, yi + hx, zi, as desired.
[email protected] MATH 532 47 From this we can get the property for rational α as follows. β We let α = γ with integer β, γ 6= 0 so that β βγhx, yi = hγx, βyi = γ2hx, yi. γ
Dividing by γ2 we get β β hx, yi = hx, yi. γ γ
Inner Product Spaces
Proof (cont.) 2 Scalar multiplication: To show hx, αyi = αhx, yi for integer α we can just repeatedly apply the additivity property just proved.
[email protected] MATH 532 48 β We let α = γ with integer β, γ 6= 0 so that β βγhx, yi = hγx, βyi = γ2hx, yi. γ
Dividing by γ2 we get β β hx, yi = hx, yi. γ γ
Inner Product Spaces
Proof (cont.) 2 Scalar multiplication: To show hx, αyi = αhx, yi for integer α we can just repeatedly apply the additivity property just proved.
From this we can get the property for rational α as follows.
[email protected] MATH 532 48 Dividing by γ2 we get β β hx, yi = hx, yi. γ γ
Inner Product Spaces
Proof (cont.) 2 Scalar multiplication: To show hx, αyi = αhx, yi for integer α we can just repeatedly apply the additivity property just proved.
From this we can get the property for rational α as follows. β We let α = γ with integer β, γ 6= 0 so that β βγhx, yi = hγx, βyi = γ2hx, yi. γ
[email protected] MATH 532 48 Inner Product Spaces
Proof (cont.) 2 Scalar multiplication: To show hx, αyi = αhx, yi for integer α we can just repeatedly apply the additivity property just proved.
From this we can get the property for rational α as follows. β We let α = γ with integer β, γ 6= 0 so that β βγhx, yi = hγx, βyi = γ2hx, yi. γ
Dividing by γ2 we get β β hx, yi = hx, yi. γ γ
[email protected] MATH 532 48 Now we take a sequence {αn} of rational numbers such that αn → α for n → ∞ and have — by continuity
hx, αnyi → hx, αyi as n → ∞.
Inner Product Spaces
Proof (cont.) Finally, for real α we use the continuity of the norm function (see HW) which implies that our inner product h·, ·i also is continuous.
[email protected] MATH 532 49 Inner Product Spaces
Proof (cont.) Finally, for real α we use the continuity of the norm function (see HW) which implies that our inner product h·, ·i also is continuous.
Now we take a sequence {αn} of rational numbers such that αn → α for n → ∞ and have — by continuity
hx, αnyi → hx, αyi as n → ∞.
[email protected] MATH 532 49 Remark Since many problems are more easily dealt with in inner product spaces (since we then have lengths and angles, see next section) the 2-norm has a clear advantage over other p-norms.
Inner Product Spaces
Theorem The only vector p-norm induced by an inner product is the 2-norm.
[email protected] MATH 532 50 Inner Product Spaces
Theorem The only vector p-norm induced by an inner product is the 2-norm.
Remark Since many problems are more easily dealt with in inner product spaces (since we then have lengths and angles, see next section) the 2-norm has a clear advantage over other p-norms.
[email protected] MATH 532 50 Therefore we need to show that it doesn’t work for p 6= 2. We do this by showing that the parallelogram identity kx + yk2 + kx − yk2 = 2 kxk2 + kyk2
fails for p 6= 2. We will do this for 1 ≤ p < ∞. You will work out the case p = ∞ in a HW problem.
Inner Product Spaces
Proof We know that the 2-norm does induce an inner product, i.e., √ T kxk2 = x x.
[email protected] MATH 532 51 We do this by showing that the parallelogram identity kx + yk2 + kx − yk2 = 2 kxk2 + kyk2
fails for p 6= 2. We will do this for 1 ≤ p < ∞. You will work out the case p = ∞ in a HW problem.
Inner Product Spaces
Proof We know that the 2-norm does induce an inner product, i.e., √ T kxk2 = x x.
Therefore we need to show that it doesn’t work for p 6= 2.
[email protected] MATH 532 51 We will do this for 1 ≤ p < ∞. You will work out the case p = ∞ in a HW problem.
Inner Product Spaces
Proof We know that the 2-norm does induce an inner product, i.e., √ T kxk2 = x x.
Therefore we need to show that it doesn’t work for p 6= 2. We do this by showing that the parallelogram identity kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 fails for p 6= 2.
[email protected] MATH 532 51 Inner Product Spaces
Proof We know that the 2-norm does induce an inner product, i.e., √ T kxk2 = x x.
Therefore we need to show that it doesn’t work for p 6= 2. We do this by showing that the parallelogram identity kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 fails for p 6= 2. We will do this for 1 ≤ p < ∞. You will work out the case p = ∞ in a HW problem.
[email protected] MATH 532 51 n !2/p X p 2/p = |[e1 + e2]i | = 2 i=1 and, similarly 2 2 2/p kx − ykp = ke1 − e2kp = 2 . Together, the left-hand side of the parallelogram identity is 2 22/p = 22/p+1.
Inner Product Spaces
Proof (cont.)
All we need is a counterexample, so we take x = e1 and y = e2 so that
2 2 kx + ykp = ke1 + e2kp
[email protected] MATH 532 52 = 22/p
and, similarly 2 2 2/p kx − ykp = ke1 − e2kp = 2 . Together, the left-hand side of the parallelogram identity is 2 22/p = 22/p+1.
Inner Product Spaces
Proof (cont.)
All we need is a counterexample, so we take x = e1 and y = e2 so that
n !2/p 2 2 X p kx + ykp = ke1 + e2kp = |[e1 + e2]i | i=1
[email protected] MATH 532 52 and, similarly 2 2 2/p kx − ykp = ke1 − e2kp = 2 . Together, the left-hand side of the parallelogram identity is 2 22/p = 22/p+1.
Inner Product Spaces
Proof (cont.)
All we need is a counterexample, so we take x = e1 and y = e2 so that
n !2/p 2 2 X p 2/p kx + ykp = ke1 + e2kp = |[e1 + e2]i | = 2 i=1
[email protected] MATH 532 52 Together, the left-hand side of the parallelogram identity is 2 22/p = 22/p+1.
Inner Product Spaces
Proof (cont.)
All we need is a counterexample, so we take x = e1 and y = e2 so that
n !2/p 2 2 X p 2/p kx + ykp = ke1 + e2kp = |[e1 + e2]i | = 2 i=1 and, similarly 2 2 2/p kx − ykp = ke1 − e2kp = 2 .
[email protected] MATH 532 52 Inner Product Spaces
Proof (cont.)
All we need is a counterexample, so we take x = e1 and y = e2 so that
n !2/p 2 2 X p 2/p kx + ykp = ke1 + e2kp = |[e1 + e2]i | = 2 i=1 and, similarly 2 2 2/p kx − ykp = ke1 − e2kp = 2 . Together, the left-hand side of the parallelogram identity is 2 22/p = 22/p+1.
[email protected] MATH 532 52 2 2 = 1 = ke2kp = kykp,
so that the right-hand side comes out to 4. Finally, we have
2 2 22/p+1 = 4 ⇐⇒ + 1 = 2 ⇐⇒ = 1 or p = 2. p p
Inner Product Spaces
Proof (cont.) For the right-hand side of the parallelogram identity we calculate
2 2 kxkp = ke1kp
[email protected] MATH 532 53 2 2 = ke2kp = kykp,
so that the right-hand side comes out to 4. Finally, we have
2 2 22/p+1 = 4 ⇐⇒ + 1 = 2 ⇐⇒ = 1 or p = 2. p p
Inner Product Spaces
Proof (cont.) For the right-hand side of the parallelogram identity we calculate
2 2 kxkp = ke1kp = 1
[email protected] MATH 532 53 4. Finally, we have
2 2 22/p+1 = 4 ⇐⇒ + 1 = 2 ⇐⇒ = 1 or p = 2. p p
Inner Product Spaces
Proof (cont.) For the right-hand side of the parallelogram identity we calculate
2 2 2 2 kxkp = ke1kp = 1 = ke2kp = kykp, so that the right-hand side comes out to
[email protected] MATH 532 53 Finally, we have
2 2 22/p+1 = 4 ⇐⇒ + 1 = 2 ⇐⇒ = 1 or p = 2. p p
Inner Product Spaces
Proof (cont.) For the right-hand side of the parallelogram identity we calculate
2 2 2 2 kxkp = ke1kp = 1 = ke2kp = kykp, so that the right-hand side comes out to 4.
[email protected] MATH 532 53 2 2 ⇐⇒ + 1 = 2 ⇐⇒ = 1 or p = 2. p p
Inner Product Spaces
Proof (cont.) For the right-hand side of the parallelogram identity we calculate
2 2 2 2 kxkp = ke1kp = 1 = ke2kp = kykp, so that the right-hand side comes out to 4. Finally, we have
22/p+1 = 4
[email protected] MATH 532 53 Inner Product Spaces
Proof (cont.) For the right-hand side of the parallelogram identity we calculate
2 2 2 2 kxkp = ke1kp = 1 = ke2kp = kykp, so that the right-hand side comes out to 4. Finally, we have
2 2 22/p+1 = 4 ⇐⇒ + 1 = 2 ⇐⇒ = 1 or p = 2. p p
[email protected] MATH 532 53 Orthogonal Vectors Outline
1 Vector Norms
2 Matrix Norms
3 Inner Product Spaces
4 Orthogonal Vectors
5 Gram–Schmidt Orthogonalization & QR Factorization
6 Unitary and Orthogonal Matrices
7 Orthogonal Reduction
8 Complementary Subspaces
9 Orthogonal Decomposition
10 Singular Value Decomposition
11 Orthogonal Projections
[email protected] MATH 532 54 Definition Two vectors x, y ∈ V are called orthogonal if
hx, yi = 0.
We often use the notation x ⊥ y.
Orthogonal Vectors Orthogonal Vectors
We will now work in a general inner product space V with induced norm k · k = ph·, ·i.
[email protected] MATH 532 55 Orthogonal Vectors Orthogonal Vectors
We will now work in a general inner product space V with induced norm k · k = ph·, ·i.
Definition Two vectors x, y ∈ V are called orthogonal if
hx, yi = 0.
We often use the notation x ⊥ y.
[email protected] MATH 532 55 Moreover, the law of cosines states
kx − yk2 = kxk2 + kyk2 − 2kxkkyk cos θ,
so that
2 2 2 T kxk + kyk − kx − yk Pythagoras 2x y cos θ = = . 2kxkkyk 2kxkkyk
This motivates our general definition of angles: Definition Let x, y ∈ V. The angle between x and y is defined via
hx, yi cos θ = , θ ∈ [0, π]. kxkkyk
Orthogonal Vectors In the HW you will prove the Pythagorean theorem for the 2-norm and standard inner product xT y, i.e., kxk2 + kyk2 = kx − yk2 ⇐⇒ xT y = 0.
[email protected] MATH 532 56 so that
2 2 2 T kxk + kyk − kx − yk Pythagoras 2x y cos θ = = . 2kxkkyk 2kxkkyk
This motivates our general definition of angles: Definition Let x, y ∈ V. The angle between x and y is defined via
hx, yi cos θ = , θ ∈ [0, π]. kxkkyk
Orthogonal Vectors In the HW you will prove the Pythagorean theorem for the 2-norm and standard inner product xT y, i.e., kxk2 + kyk2 = kx − yk2 ⇐⇒ xT y = 0. Moreover, the law of cosines states
kx − yk2 = kxk2 + kyk2 − 2kxkkyk cos θ,
[email protected] MATH 532 56 This motivates our general definition of angles: Definition Let x, y ∈ V. The angle between x and y is defined via
hx, yi cos θ = , θ ∈ [0, π]. kxkkyk
Orthogonal Vectors In the HW you will prove the Pythagorean theorem for the 2-norm and standard inner product xT y, i.e., kxk2 + kyk2 = kx − yk2 ⇐⇒ xT y = 0. Moreover, the law of cosines states
kx − yk2 = kxk2 + kyk2 − 2kxkkyk cos θ, so that
2 2 2 T kxk + kyk − kx − yk Pythagoras 2x y cos θ = = . 2kxkkyk 2kxkkyk
[email protected] MATH 532 56 Orthogonal Vectors In the HW you will prove the Pythagorean theorem for the 2-norm and standard inner product xT y, i.e., kxk2 + kyk2 = kx − yk2 ⇐⇒ xT y = 0. Moreover, the law of cosines states
kx − yk2 = kxk2 + kyk2 − 2kxkkyk cos θ, so that
2 2 2 T kxk + kyk − kx − yk Pythagoras 2x y cos θ = = . 2kxkkyk 2kxkkyk
This motivates our general definition of angles: Definition Let x, y ∈ V. The angle between x and y is defined via
hx, yi cos θ = , θ ∈ [0, π]. kxkkyk
[email protected] MATH 532 56 Theorem Every orthonormal set is linearly independent.
Corollary Every orthonormal set of n vectors from an n-dimensional vector space V is an orthonormal basis for V.
Orthogonal Vectors Orthonormal sets
Definition
A set {u1, u2,..., un} ⊆ V is called orthonormal if
hui , uj i = δij (Kronecker delta).
[email protected] MATH 532 57 Corollary Every orthonormal set of n vectors from an n-dimensional vector space V is an orthonormal basis for V.
Orthogonal Vectors Orthonormal sets
Definition
A set {u1, u2,..., un} ⊆ V is called orthonormal if
hui , uj i = δij (Kronecker delta).
Theorem Every orthonormal set is linearly independent.
[email protected] MATH 532 57 Orthogonal Vectors Orthonormal sets
Definition
A set {u1, u2,..., un} ⊆ V is called orthonormal if
hui , uj i = δij (Kronecker delta).
Theorem Every orthonormal set is linearly independent.
Corollary Every orthonormal set of n vectors from an n-dimensional vector space V is an orthonormal basis for V.
[email protected] MATH 532 57 n X ⇐⇒ αj hui , uj i = 0 ⇐⇒ αi = 0. j=1 | {z } =δij
Since i was arbitrary this holds for all i = 1,..., n, and we have linear independence.
To see this is true we take the inner product with ui :
n X hui , αj uj i = hui , 0i j=1
Orthogonal Vectors
Proof (of the theorem) We want to show linear independence, i.e., that
n X αj uj = 0 =⇒ αj = 0, j = 1,..., n. j=1
[email protected] MATH 532 58 n X ⇐⇒ αj hui , uj i = 0 ⇐⇒ αi = 0. j=1 | {z } =δij
Since i was arbitrary this holds for all i = 1,..., n, and we have linear independence.
Orthogonal Vectors
Proof (of the theorem) We want to show linear independence, i.e., that
n X αj uj = 0 =⇒ αj = 0, j = 1,..., n. j=1
To see this is true we take the inner product with ui :
n X hui , αj uj i = hui , 0i j=1
[email protected] MATH 532 58 Since i was arbitrary this holds for all i = 1,..., n, and we have linear independence.
Orthogonal Vectors
Proof (of the theorem) We want to show linear independence, i.e., that
n X αj uj = 0 =⇒ αj = 0, j = 1,..., n. j=1
To see this is true we take the inner product with ui :
n X hui , αj uj i = hui , 0i j=1 n X ⇐⇒ αj hui , uj i = 0 ⇐⇒ αi = 0. j=1 | {z } =δij
[email protected] MATH 532 58 Orthogonal Vectors
Proof (of the theorem) We want to show linear independence, i.e., that
n X αj uj = 0 =⇒ αj = 0, j = 1,..., n. j=1
To see this is true we take the inner product with ui :
n X hui , αj uj i = hui , 0i j=1 n X ⇐⇒ αj hui , uj i = 0 ⇐⇒ αi = 0. j=1 | {z } =δij
Since i was arbitrary this holds for all i = 1,..., n, and we have linear independence.
[email protected] MATH 532 58 Using this basis we can express any x ∈ Rn as
x = x1e1 + x2e2 + ... + xnen,
we get a coordinate expansion of x.
Orthogonal Vectors
Example The standard orthonormal basis of Rn is given by
{e1, e2,..., en}.
[email protected] MATH 532 59 Orthogonal Vectors
Example The standard orthonormal basis of Rn is given by
{e1, e2,..., en}.
Using this basis we can express any x ∈ Rn as
x = x1e1 + x2e2 + ... + xnen, we get a coordinate expansion of x.
[email protected] MATH 532 59 Consider the orthonormal basis B = {u1, u2,..., un} and assume
n X x = αj uj j=1
for some appropriate scalars αj . To find these expansion coefficients αj we take the inner product with ui , i.e., n X hui , xi = αj hui , uj i = αi . j=1 | {z } =δij
Orthogonal Vectors
In fact, any other orthonormal basis provides just as simple a representation of x;
[email protected] MATH 532 60 To find these expansion coefficients αj we take the inner product with ui , i.e., n X hui , xi = αj hui , uj i = αi . j=1 | {z } =δij
Orthogonal Vectors
In fact, any other orthonormal basis provides just as simple a representation of x;
Consider the orthonormal basis B = {u1, u2,..., un} and assume
n X x = αj uj j=1 for some appropriate scalars αj .
[email protected] MATH 532 60 n X = αj hui , uj i = αi . j=1 | {z } =δij
Orthogonal Vectors
In fact, any other orthonormal basis provides just as simple a representation of x;
Consider the orthonormal basis B = {u1, u2,..., un} and assume
n X x = αj uj j=1 for some appropriate scalars αj . To find these expansion coefficients αj we take the inner product with ui , i.e.,
hui , xi
[email protected] MATH 532 60 = αi .
Orthogonal Vectors
In fact, any other orthonormal basis provides just as simple a representation of x;
Consider the orthonormal basis B = {u1, u2,..., un} and assume
n X x = αj uj j=1 for some appropriate scalars αj . To find these expansion coefficients αj we take the inner product with ui , i.e., n X hui , xi = αj hui , uj i j=1 | {z } =δij
[email protected] MATH 532 60 Orthogonal Vectors
In fact, any other orthonormal basis provides just as simple a representation of x;
Consider the orthonormal basis B = {u1, u2,..., un} and assume
n X x = αj uj j=1 for some appropriate scalars αj . To find these expansion coefficients αj we take the inner product with ui , i.e., n X hui , xi = αj hui , uj i = αi . j=1 | {z } =δij
[email protected] MATH 532 60 Orthogonal Vectors
We therefore have proved Theorem
Let {u1, u2,..., un} be an orthonormal basis for an inner product space V. Then any x ∈ V can be written as
n X x = hx, ui iui . j=1
This is a (finite) Fourier expansion with Fourier coefficients hx, ui i.
[email protected] MATH 532 61 Orthogonal Vectors
Remark The classical (infinite-dimensional) Fourier series for continuous functions on (−π, π) uses the orthogonal (but not yet orthonormal) basis {1, sin t, cos t, sin 2t, cos 2t,..., } and the inner product Z π hf , gi = f (t)g(t)dt. −π
[email protected] MATH 532 62 It is clear by inspection that B is an orthogonal subset of R3, i.e., using T the Euclidean inner product, we have ui uj = 0, i, j = 1, 2, 3, i 6= j. We can obtain an orthonormal basis by normalizing the vectors, i.e., by ui computing v i = , i = 1, 2, 3. kui k2 This yields
1 0 1 1 1 v 1 = √ 0 , v 2 = 1 , v 3 = √ 0 . 2 1 0 2 −1
Orthogonal Vectors
Example Consider the basis 1 0 1 B = {u1, u2, u3} = 0 , 1 , 0 . 1 0 −1
[email protected] MATH 532 63 We can obtain an orthonormal basis by normalizing the vectors, i.e., by ui computing v i = , i = 1, 2, 3. kui k2 This yields
1 0 1 1 1 v 1 = √ 0 , v 2 = 1 , v 3 = √ 0 . 2 1 0 2 −1
Orthogonal Vectors
Example Consider the basis 1 0 1 B = {u1, u2, u3} = 0 , 1 , 0 . 1 0 −1
It is clear by inspection that B is an orthogonal subset of R3, i.e., using T the Euclidean inner product, we have ui uj = 0, i, j = 1, 2, 3, i 6= j.
[email protected] MATH 532 63 This yields
1 0 1 1 1 v 1 = √ 0 , v 2 = 1 , v 3 = √ 0 . 2 1 0 2 −1
Orthogonal Vectors
Example Consider the basis 1 0 1 B = {u1, u2, u3} = 0 , 1 , 0 . 1 0 −1
It is clear by inspection that B is an orthogonal subset of R3, i.e., using T the Euclidean inner product, we have ui uj = 0, i, j = 1, 2, 3, i 6= j. We can obtain an orthonormal basis by normalizing the vectors, i.e., by ui computing v i = , i = 1, 2, 3. kui k2
[email protected] MATH 532 63 Orthogonal Vectors
Example Consider the basis 1 0 1 B = {u1, u2, u3} = 0 , 1 , 0 . 1 0 −1
It is clear by inspection that B is an orthogonal subset of R3, i.e., using T the Euclidean inner product, we have ui uj = 0, i, j = 1, 2, 3, i 6= j. We can obtain an orthonormal basis by normalizing the vectors, i.e., by ui computing v i = , i = 1, 2, 3. kui k2 This yields
1 0 1 1 1 v 1 = √ 0 , v 2 = 1 , v 3 = √ 0 . 2 1 0 2 −1
[email protected] MATH 532 63 1 0 1 4 1 2 1 = √ √ 0 + 2 1 − √ √ 0 2 2 1 0 2 2 −1 2 0 −1 = 0 + 2 + 0 . 2 0 1
3 X T x = x v i v i i=1
Orthogonal Vectors
Example (cont.) T The Fourier expansion of x = 1 2 3 is given by
[email protected] MATH 532 64 1 0 1 4 1 2 1 = √ √ 0 + 2 1 − √ √ 0 2 2 1 0 2 2 −1 2 0 −1 = 0 + 2 + 0 . 2 0 1
Orthogonal Vectors
Example (cont.) T The Fourier expansion of x = 1 2 3 is given by
3 X T x = x v i v i i=1
[email protected] MATH 532 64 2 0 −1 = 0 + 2 + 0 . 2 0 1
Orthogonal Vectors
Example (cont.) T The Fourier expansion of x = 1 2 3 is given by
3 X T x = x v i v i i=1 1 0 1 4 1 2 1 = √ √ 0 + 2 1 − √ √ 0 2 2 1 0 2 2 −1
[email protected] MATH 532 64 Orthogonal Vectors
Example (cont.) T The Fourier expansion of x = 1 2 3 is given by
3 X T x = x v i v i i=1 1 0 1 4 1 2 1 = √ √ 0 + 2 1 − √ √ 0 2 2 1 0 2 2 −1 2 0 −1 = 0 + 2 + 0 . 2 0 1
[email protected] MATH 532 64 Gram–Schmidt Orthogonalization & QR Factorization Outline
1 Vector Norms
2 Matrix Norms
3 Inner Product Spaces
4 Orthogonal Vectors
5 Gram–Schmidt Orthogonalization & QR Factorization
6 Unitary and Orthogonal Matrices
7 Orthogonal Reduction
8 Complementary Subspaces
9 Orthogonal Decomposition
10 Singular Value Decomposition
11 Orthogonal Projections
[email protected] MATH 532 65 Idea: construct u1, u2,..., un successively so that {u1, u2,..., uk } is an ON basis for span{x1, x2,..., xk }, k = 1,..., n.
Gram–Schmidt Orthogonalization & QR Factorization
We want to convert an arbitrary basis {x1, x2,..., xn} of V to an orthonormal basis {u1, u2,..., un}.
[email protected] MATH 532 66 Gram–Schmidt Orthogonalization & QR Factorization
We want to convert an arbitrary basis {x1, x2,..., xn} of V to an orthonormal basis {u1, u2,..., un}.
Idea: construct u1, u2,..., un successively so that {u1, u2,..., uk } is an ON basis for span{x1, x2,..., xk }, k = 1,..., n.
[email protected] MATH 532 66 k = 2: Consider the projection of x2 onto u1, i.e.,
hu1, x2iu1.
Then v 2 = x2 − hu1, x2iu1 and v 2 u2 = . kv 2k
Gram–Schmidt Orthogonalization & QR Factorization Construction
k = 1: x1 u1 = . kx1k
[email protected] MATH 532 67 Then v 2 = x2 − hu1, x2iu1 and v 2 u2 = . kv 2k
Gram–Schmidt Orthogonalization & QR Factorization Construction
k = 1: x1 u1 = . kx1k
k = 2: Consider the projection of x2 onto u1, i.e.,
hu1, x2iu1.
[email protected] MATH 532 67 Gram–Schmidt Orthogonalization & QR Factorization Construction
k = 1: x1 u1 = . kx1k
k = 2: Consider the projection of x2 onto u1, i.e.,
hu1, x2iu1.
Then v 2 = x2 − hu1, x2iu1 and v 2 u2 = . kv 2k
[email protected] MATH 532 67 k+1 X xk+1 = hui , xk+1iui i=1 k X ⇐⇒ xk+1 = hui , xk+1iui + huk+1, xk+1iuk+1 i=1 Pk xk+1 − i=1hui , xk+1iui v k+1 ⇐⇒ uk+1 = = huk+1, xk+1i huk+1, xk+1i This vector, however is not yet normalized.
Use the Fourier expansion to express xk+1 with respect to {u1,..., uk+1}:
Gram–Schmidt Orthogonalization & QR Factorization
In general, consider {u1,..., uk } as a given ON basis for span{x1,..., xk }.
[email protected] MATH 532 68 k+1 X xk+1 = hui , xk+1iui i=1 k X ⇐⇒ xk+1 = hui , xk+1iui + huk+1, xk+1iuk+1 i=1 Pk xk+1 − i=1hui , xk+1iui v k+1 ⇐⇒ uk+1 = = huk+1, xk+1i huk+1, xk+1i This vector, however is not yet normalized.
Gram–Schmidt Orthogonalization & QR Factorization
In general, consider {u1,..., uk } as a given ON basis for span{x1,..., xk }. Use the Fourier expansion to express xk+1 with respect to {u1,..., uk+1}:
[email protected] MATH 532 68 k X ⇐⇒ xk+1 = hui , xk+1iui + huk+1, xk+1iuk+1 i=1 Pk xk+1 − i=1hui , xk+1iui v k+1 ⇐⇒ uk+1 = = huk+1, xk+1i huk+1, xk+1i This vector, however is not yet normalized.
Gram–Schmidt Orthogonalization & QR Factorization
In general, consider {u1,..., uk } as a given ON basis for span{x1,..., xk }. Use the Fourier expansion to express xk+1 with respect to {u1,..., uk+1}:
k+1 X xk+1 = hui , xk+1iui i=1
[email protected] MATH 532 68 Pk xk+1 − i=1hui , xk+1iui v k+1 ⇐⇒ uk+1 = = huk+1, xk+1i huk+1, xk+1i This vector, however is not yet normalized.
Gram–Schmidt Orthogonalization & QR Factorization
In general, consider {u1,..., uk } as a given ON basis for span{x1,..., xk }. Use the Fourier expansion to express xk+1 with respect to {u1,..., uk+1}:
k+1 X xk+1 = hui , xk+1iui i=1 k X ⇐⇒ xk+1 = hui , xk+1iui + huk+1, xk+1iuk+1 i=1
[email protected] MATH 532 68 v = k+1 huk+1, xk+1i This vector, however is not yet normalized.
Gram–Schmidt Orthogonalization & QR Factorization
In general, consider {u1,..., uk } as a given ON basis for span{x1,..., xk }. Use the Fourier expansion to express xk+1 with respect to {u1,..., uk+1}:
k+1 X xk+1 = hui , xk+1iui i=1 k X ⇐⇒ xk+1 = hui , xk+1iui + huk+1, xk+1iuk+1 i=1 Pk xk+1 − i=1hui , xk+1iui ⇐⇒ uk+1 = huk+1, xk+1i
[email protected] MATH 532 68 This vector, however is not yet normalized.
Gram–Schmidt Orthogonalization & QR Factorization
In general, consider {u1,..., uk } as a given ON basis for span{x1,..., xk }. Use the Fourier expansion to express xk+1 with respect to {u1,..., uk+1}:
k+1 X xk+1 = hui , xk+1iui i=1 k X ⇐⇒ xk+1 = hui , xk+1iui + huk+1, xk+1iuk+1 i=1 Pk xk+1 − i=1hui , xk+1iui v k+1 ⇐⇒ uk+1 = = huk+1, xk+1i huk+1, xk+1i
[email protected] MATH 532 68 Gram–Schmidt Orthogonalization & QR Factorization
In general, consider {u1,..., uk } as a given ON basis for span{x1,..., xk }. Use the Fourier expansion to express xk+1 with respect to {u1,..., uk+1}:
k+1 X xk+1 = hui , xk+1iui i=1 k X ⇐⇒ xk+1 = hui , xk+1iui + huk+1, xk+1iuk+1 i=1 Pk xk+1 − i=1hui , xk+1iui v k+1 ⇐⇒ uk+1 = = huk+1, xk+1i huk+1, xk+1i This vector, however is not yet normalized.
[email protected] MATH 532 68 k X =⇒ kv k+1k = kxk+1 − hui , xk+1iui k = |huk+1, xk+1i|. i=1 Therefore
k X huk+1, xk+1i = ±kxk+1 − hui , xk+1iui k. i=1 Since the factor ±1 does not change the span, nor orthogonality, nor normalization we can pick the positive sign.
Gram–Schmidt Orthogonalization & QR Factorization
We now want kuk+1k = 1, i.e., r v k+1 v k+1 1 h , i = kv k+1k = 1 huk+1, xk+1i huk+1, xk+1i |huk+1, xk+1i|
[email protected] MATH 532 69 Therefore
k X huk+1, xk+1i = ±kxk+1 − hui , xk+1iui k. i=1 Since the factor ±1 does not change the span, nor orthogonality, nor normalization we can pick the positive sign.
Gram–Schmidt Orthogonalization & QR Factorization
We now want kuk+1k = 1, i.e., r v k+1 v k+1 1 h , i = kv k+1k = 1 huk+1, xk+1i huk+1, xk+1i |huk+1, xk+1i| k X =⇒ kv k+1k = kxk+1 − hui , xk+1iui k = |huk+1, xk+1i|. i=1
[email protected] MATH 532 69 Since the factor ±1 does not change the span, nor orthogonality, nor normalization we can pick the positive sign.
Gram–Schmidt Orthogonalization & QR Factorization
We now want kuk+1k = 1, i.e., r v k+1 v k+1 1 h , i = kv k+1k = 1 huk+1, xk+1i huk+1, xk+1i |huk+1, xk+1i| k X =⇒ kv k+1k = kxk+1 − hui , xk+1iui k = |huk+1, xk+1i|. i=1 Therefore
k X huk+1, xk+1i = ±kxk+1 − hui , xk+1iui k. i=1
[email protected] MATH 532 69 Gram–Schmidt Orthogonalization & QR Factorization
We now want kuk+1k = 1, i.e., r v k+1 v k+1 1 h , i = kv k+1k = 1 huk+1, xk+1i huk+1, xk+1i |huk+1, xk+1i| k X =⇒ kv k+1k = kxk+1 − hui , xk+1iui k = |huk+1, xk+1i|. i=1 Therefore
k X huk+1, xk+1i = ±kxk+1 − hui , xk+1iui k. i=1 Since the factor ±1 does not change the span, nor orthogonality, nor normalization we can pick the positive sign.
[email protected] MATH 532 69 Gram–Schmidt Orthogonalization & QR Factorization Gram–Schmidt algorithm
Summarizing, we have
x1 u1 = , kx1k k−1 X v k = xk − hui , xk iui , k = 2,..., n, i=1 v k uk = . kv k k
[email protected] MATH 532 70 Gram–Schmidt Orthogonalization & QR Factorization Using matrix notation to describe Gram–Schmidt
We will assume V ⊆ Rm (but this also works in the complex case). Let 0 . m U1 = . ∈ R 0 and for k = 2, 3,..., n let