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Numerical Linear Algebra Lecture 3 Numerical Linear Algebra Lecture 3 Larisa Beilina Lecture 3 NLA notions Norms Sherman-Morrison formula Perturbation theory Condition number, relative condition number Gaussian elimination Larisa Beilina Lecture 3 Singular values The singular values of a compact operator T : X Y acting between Hilbert spaces X and Y , are the square roots of the→ eigenvalues of the nonnegative self-adjoint operator T ∗T : X X (where T ∗ denotes the adjoint of T ). For the case of a matrix A, singular→ values are computed as: σ = λ(A∗A). The singularp values are nonnegative real numbers, usually listed in decreasing order (σ1(T ),σ2(T ),...). If T is self-adjoint, then the largest singular value σ1(T ) is equal to the operator norm of T . In the case of a normal matrix A (or A∗A = AA∗, when A is real then AT A = AAT ), the spectral theorem can be applied to obtain unitary diagonalization of A as A = UΛU∗. Therefore, √A∗A = U Λ U∗ and so the singular values are simply the absolute values of the eig|envalues.| Larisa Beilina Lecture 3 Norm of a vector We are going to use following such called Lp-norms which are usually denoted by p: k · k n p 1/p x p =( xk ) , p 1. k k | | ≥ Xk=1 n p = 1, x 1 = xk , one-norm k k k=1 | | P n 2 1/2 p = 2, x 2 =( xk ) , two-norm k k k=1 | | P p = , x = max1 k n xk , max-norm or infinity-norm. ∞ k k∞ ≤ ≤ | | Larisa Beilina Lecture 3 Norm of a vector All these norms, as all other vector norms, has following properties: x = 0 x > 0 (positivity), 0 = 0. 6 → k k k k αx = α x for all α R (homogeneity) k k | |k k ∈ x + y x + y (triangle inequality) k k ≤ k k k k Larisa Beilina Lecture 3 Norm of a vector Norms are defined differently, but they can be compared. Two norms α and β on a vector space V are called equivalent if there exist positivek · k realk · numbers k C and D such that for all x in V C x x D x . k kα ≤ k kβ ≤ k kα T For example, let x =(x1, ...., xn) α x 1 x 2 β x 1 k k ≤ k k ≤ k k In this case we have: α = 1/√n,β = 1. Larisa Beilina Lecture 3 Norm of a vector Other examples: x 2 x 1 √n x 2 k k ≤ k k ≤ k k x x 2 √n x k k∞ ≤ k k ≤ k k∞ x x 1 n x , k k∞ ≤ k k ≤ k k∞ x x 2 x 1 √n x 2 n x k k∞ ≤ k k ≤ k k ≤ k k ≤ k k∞ If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent. Larisa Beilina Lecture 3 Inner product The dot product of two vectors x =(x1,..., xn) and y =(y1,..., yn) is defined as: n T T x y =(x, y)= xk yk = x y, x 2 = √x x. · k=1 k k We note that xPT y is a scalar, but xy T is a matrix. Example 3 x T y =[ 1, 2, 3] 2 =( 1) (3)+ 2 2 + 3 1 = 4. − · 1 − · · · 1 3 2 1 T − − − − xy = 2 [3, 2, 1]= 6 4 2 3 · 9 6 3 Larisa Beilina Lecture 3 Norm of a vector Example 1 − x 1 = 1 + 2 + 3 + 5 = 11 2 x = ; || || |− | | | | | |− | 3 2 2 2 2 x 2 = ( 1) + 2 + 3 +( 5) = √39 5 || || − − − p x = max( 1 , 2 , 3 , 5 )= 5. || ||∞ |− | | | | | |− | Larisa Beilina Lecture 3 Norm of a vector x A vector x is normalized if x = 1. If x = 0 then is normalized || || 6 x vector. || || Example 1 − T 2 2 x x =[ 1, 2, 3, 5] =( 1) ( 1)+ 2 2 + 3 3 +( 5) = 39 − − · 3 − · − · · − 5 − x 2 = √39 || || x 1 2 3 5 T V = = − , , , − = V 2 = 1 x √39 √39 √39 √39 ⇒ || || || || Larisa Beilina Lecture 3 Matrix norm. Definition Let K will denote the field of real or complex numbers. Let K m×n denote the vector space containing all matrices with m rows and n columns with entries in K. Let A∗ denotes the conjugate transpose of matrix A. A matrix norm is a vector norm on K m×n. That is, if A denotes the norm of the matrix A, then, k k A > 0 if A = 0 and A = 0 if A = 0. k k 6 k k αA = α A for all α in K and all matrices A in K m×n. k k | |k k A + B A + B for all matrices A and B in K m×n. k k ≤ k k k k Larisa Beilina Lecture 3 Matrix norm. Definition In the case of square matrices (thus, m = n), some (but not all) matrix norms satisfy the following condition, which is related to the fact that matrices are more than just vectors: AB A B for all matrices A and B in K n×n. Ak matrixk ≤ k normkk thatk satisfies this additional property is called a sub-multiplicative norm. The set of all n-by-n matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra. Larisa Beilina Lecture 3 Matrix norm. Induced norm If vector norms on Km and Kn are given (K is field of real or complex numbers), then one defines the corresponding induced norm or operator norm on the space of m-by-n matrices as the following maxima: A = max Ax : x K n with x = 1 k k {k k ∈ k k } Ax = max k k : x K n with x = 0 . x ∈ 6 k k If m = n and one uses the same norm on the domain and the range, then the induced operator norm is a sub-multiplicative matrix norm. Larisa Beilina Lecture 3 Matrix norm. p-norm The operator norm corresponding to the p-norm for vectors is: Ax p A p = max k k . k k x=0 x 6 k kp In the case of p = 1 and p = , the norms can be computed as: ∞ m A 1 = max aij , k k 1 j n | | ≤ ≤ Xi=1 which is simply the maximum absolute column sum of the matrix. n A = max aij , k k∞ 1 i m | | ≤ ≤ Xj=1 which is simply the maximum absolute row sum of the matrix Larisa Beilina Lecture 3 Matrix norm. Example For example, if the matrix A is defined by 3 5 7 A = 2 6 4 , 0 2 8 then we have A 1 = max(5, 13, 19)= 19 and A = max(||15||, 12, 10)= 15. Consider another example || ||∞ 2 4 2 1 3 1 5 2 A = , 1 2 3 3 0 6 1 2 where we add all the entries in each column and determine the greatest value, which results in A 1 = max(6, 13, 11, 8)= 13. || || We can do the same for the rows and get A ∞ = max(9, 11, 9, 9)= 11. Thus 11 is our max. || || Larisa Beilina Lecture 3 Matrix norm. Example In the special case of p = 2 (the Euclidean norm) and m = n (square matrices), the induced matrix norm is the spectral norm. The spectral norm of a matrix A is the largest singular value of A i.e. the square root of the largest eigenvalue of the positive-semidefinite matrix A∗A: ∗ A 2 = λmax(A A)= σmax(A) k k p where A∗ denotes the conjugate transpose of A. Larisa Beilina Lecture 3 Matrix norm. Example Example 1 2 3 − − A = 6 4 2 9 6 3 − A 2 = max λ(AT A) || || p 1 6 9 118 32 36 T T − A = 2 4 6 , A A = 32 56 4 −3 2− 3 −36 4− 22 − − 8.9683 max λ(AT A) =max(2.9947, 6.7322, λ(AT A)= 45.3229 ; p 11.9042)= 11.9042 141.7089 Larisa Beilina Lecture 3 Matrix norm. Example Example 1 0 1 0 1 λ 0 A = ; AT = ; AT A λI = − = 0; 0 1 0 1 − 0 1 λ − λ1 = 1, λ2 = 1; A 2 = max λ(AT A) = max(1, 1)= 1 || || p Larisa Beilina Lecture 3 Matrix norm Any induced norm satisfies the inequality A ρ(A), k k≥ where ρ(A) := max λ1 ,..., λm is the spectral radius of A. For a symmetric or hermitian{| | matrix| A|}, we have equality for the 2-norm, since in this case the 2-norm is the spectral radius of A. For an arbitrary matrix, we may not have equality for any norm. Example Take 0 1 A = , 0 0 the spectral radius ρ(A) of A is 0, but A is not the zero matrix, and so none of the induced norms are equal to the spectral radius of A: A 1 = 1, A = 1, A 2 = 1. k k k k∞ k k ∗ ∗ 0 0 0 1 0 0 A = λmax(A A)= σmax(A); A A = = 2 1 00 0 0 1 k k p Larisa Beilina Lecture 3 Matrix norm. "Entrywise" norms For square matrices we have the spectral radius formula: 1 lim Ar /r = ρ(A). r →∞ k k These vector norms treat an m n matrix as a vector of size m n , × · and use one of the familiar vector norms.
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