Ch 6: Eigenvalues 6.4 Hermitian Matrices We consider matrices with complex entries (ai;j 2 C) versus real entries (ai;j 2 R). 1. in R the length of a real number x is jxj = the length from the origin to the number (either in the positive or the negative direction) p 2. in C the length of a complex number z = a + bi is jzj = 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 p 1 2 n p T the norm jjzjj = (¯z z) = z¯1z1 +z ¯2z2 + ::: +z ¯nzn, which is the square root of the T inner product, thus a real number. Herep ¯z = (z ¯1; z¯2;:::; z¯n) is the conjugate of z. We denote by zH = ¯zT, and so jjzjj = 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 fw1; w2 :::; wng, 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. 22 − i 1 −2i 3 22 + i 1 +2i 3T 22 + i −5i 0 3 H For example, if A = 4 5i 0 5 − i5 then A = 4 −5i 0 5 + i5 = 4 1 0 5 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 jjxjj2 = hx; xi = xTx jjzjj2 = 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 2 R) (αA + βB) =αA ¯ + βB¯ (α; β 2 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 2 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 2 R; 8i A = Hermitian =) λi 2 R; 8i 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; 8i) =) 9U = unitary and it diagonalizes A =) 9X = 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 9U 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 9U a unitary matrix such that T = U H AU is upper triangular matrix. (c) Spectral Theorem: If A is Hermitian, then 9U a unitary matrix such that U H AU is a diagonal matrix. Note that if some eigenvalue λj has algebraic multiplicity ≥ 2, then the eigen- vectors 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 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 jjxjj2 = hx; xi = xTx jjzjj2 = 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 2 R) (αA + βB) =αA ¯ + βB¯ (α; β 2 C) (AB)T = BT AT (AB)H = BH AH if AT = A () A = symmetric if AH = A () A = 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 2 R; 8i A = Hermitian =) λi 2 R; 8i 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; 8i) =) 9U = unitary and it diagonalizes A =) 9X = 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 3.