6.4 Hermitian Matrices

Ch 6: Eigenvalues

We consider matrices with complex entries (ai,j ∈ C) versus real entries (ai,j ∈ R). 1. in R the length of a real number x is |x| = the length from the origin to the number (either in the positive or the negative direction) √ 2. in C the length of a complex number z = a + bi is |z| = a2 + b2 = the length of the a 2 2 2 vector [ b ] (from the origin to the point (a, b)). Also z = a + b . 3. in C the scalars are complex numbers z, addition of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i and product of complex numbers: (a + bi)(c + di) = (ac − bd) + (ad + bc)i, and C with + and · is a vector space. 4. Particularly is a normed vector space with the vectors z = (z , z , . . . , z )T , and C √ 1 2 n p T the norm ||z|| = (¯z z) = z¯1z1 +z ¯2z2 + ... +z ¯nzn, which is the square root of the T inner product, thus a real number. Here√ ¯z = (z ¯1, z¯2,..., z¯n) is the conjugate of z. We denote by zH = ¯zT, and so ||z|| = zHz 5. the inner product of z and w is the complex number hz, wi = wHz

6. if z is a vector in the complex vector space with the orthonormal basis {w1, w2 ..., wn}, Pn then we can write z as a linear combination of the vector basis as z = i=1hz, wiiwi (see Exercise 2 of Homework)

m×n H 7. if A is a matrix in C , then A is the matrix whose every entry is the conjugate of the corresponding entry of A. 2 − i 1 −2i  2 + i 1 +2i T 2 + i −5i 0  H For example, if A =  5i 0 5 − i then A =  −5i 0 5 + i =  1 0 5  0 5 −5i 0 5 5i 2i 5 5i n n n n 8. vectors in R versus in C and matrices in R versus in C :

n n R C

hx, yi = yTx hz, wi = wHz hx, xi ≥ 0 with equality iﬀ x = 0 hz, zi ≥ 0 with equality iﬀ z = 0 hx, yi = hy, xi hz, wi = hw, zi hαx + βy, zi = αhx, zi + βhy, zi hαz + βu, wi = αhz, wi + βhu, wi ||x||2 = hx, xi = xTx ||z||2 = hz, zi = zHz

n×n n×n R C

(AT )T = A (AH )H = A T T T H H H (αA + βB) = αA + βA (α, β ∈ R) (αA + βB) =αA ¯ + βB¯ (α, β ∈ C) (AB)T = BT AT (AB)H = BH AH if AT = A ⇐⇒ A = symmetric if AH = A ⇐⇒ A = Hermitian

1 H 9. A matrix A is a Hermitian matrix if A = A (they are ideal matrices in C since properties that one would expect for matrices will probably hold). 1 2 − i 1 2 + i For example A = is Hermitian since A¯ = and so 2 + i 0 2 − i 0 1 2 − i AH = A¯T = = A 2 + i 0 10. if A is Hermitian, then A is symmetric. However the converse fails, and here is a 1 2 − i counterexample: A = . However if A ∈ n×n is symmetric, then it is 2 − i 0 R Hermitian.

n×n n×n Symmetric and orthogonal matrices in R Hermitian and unitary matrices in C

Defn: if AT = A ⇐⇒ A = symmetric Defn: if AH = A ⇐⇒ A = Hermitian A = symmetric =⇒ A is a square matrix A = Hermitian =⇒ A is a square matrix a pure complex matrix cannot be Hermitian (the diagonal must have real entries) A = symmetric =⇒ λi ∈ R, ∀i A = Hermitian =⇒ λi ∈ R, ∀i A = symmetric =⇒ eigenvectors A = Hermitian =⇒ eigenvectors belonging to distinct eigenvalues are orthogonal belonging to distinct eigenvalues are orthogonal (see #5 page 353) Q is orthogonal if its column vectors U is unitary if its column vectors form an orthonormal set form an orthonormal set (i.e. QT Q = I = QQT ) (i.e. U H U = I = UU H )

Q is orthogonal =⇒ Q−1 = QT U is unitary =⇒ U −1 = U H if the eigenvalues of matrix A are all distinct, if A is an Hermitian matrix A, (or algebraic multipl λi = geom multipl λi, ∀i) =⇒ ∃U = unitary and it diagonalizes A =⇒ ∃X = nonsingular and it diagonalizes A (i.e. the diagonal matrix T is (i.e. the diagonal matrix D is T = U H AU or A = UTU H ) D = X−1AX or A = XDX−1) T is ﬁrst shown to be upper triangular in Thm 6.4.3 (see Remark 3 page 308) and then that it is shown to be diagonal in Thm 6.4.4

11. These three give the last row in the table above:

(a) if the eigenvalues of an Hermitian matrix A are all distinct, then ∃U that is unitary and it diagonalizes A. In this case U has as columns the normalized eigenvectors of A (b) Schur’s Theorem: If A is n × n, then ∃U a unitary matrix such that T = U H AU is upper triangular matrix. (c) Spectral Theorem: If A is Hermitian, then ∃U a unitary matrix such that U H AU is a diagonal matrix. Note that if some eigenvalue λj has algebraic multiplicity ≥ 2, then the eigenvectors corresponding to λj are not orthonormal, and so we use Gram-Schmidt to normalize them (we use Gram Schmidt for each set of eigenvectors that cor- respond to each repeated eigenvalue)

2 n n R C

n×n n×n R C

n×n n×n Symmetric and orthogonal matrices in R Hermitian and unitary matrices in C