Math 408 Advanced Linear Algebra Chi-Kwong Li Chapter 2 Unitary similarity and normal matrices n ∗ Definition A set of vector fx1; : : : ; xkg in C is orthogonal if xi xj = 0 for i 6= j. If, in addition, ∗ that xj xj = 1 for each j, then the set is orthonormal. Theorem An orthonormal set of vectors in Cn is linearly independent. n Definition An orthonormal set fx1; : : : ; xng in C is an orthonormal basis. A matrix U 2 Mn ∗ is unitary if it has orthonormal columns, i.e., U U = In. Theorem Let U 2 Mn. The following are equivalent. (a) U is unitary. (b) U is invertible and U ∗ = U −1. ∗ ∗ (c) UU = In, i.e., U is unitary. (d) kUxk = kxk for all x 2 Cn, where kxk = (x∗x)1=2. Remarks (1) Real unitary matrices are orthogonal matrices. (2) Unitary (real orthogonal) matrices form a group in the set of complex (real) matrices and is a subgroup of the group of invertible matrices. ∗ Definition Two matrices A; B 2 Mn are unitarily similar if A = U BU for some unitary U. Theorem If A; B 2 Mn are unitarily similar, then X 2 ∗ ∗ X 2 jaijj = tr (A A) = tr (B B) = jbijj : i;j i;j Specht's Theorem Two matrices A and B are similar if and only if tr W (A; A∗) = tr W (B; B∗) for all words of length of degree at most 2n2. Schur's Theorem Every A 2 Mn is unitarily similar to a upper triangular matrix. Theorem Every A 2 Mn(R) is orthogonally similar to a matrix in upper triangular block form so that the diagonal blocks have size at most 2. Corollary Let A 2 Mn and " > 0. (a) There is A~ 2 Mn with distinct eigenvalues such that kA − A~kF < ", where kXkF = (tr X∗X)1=2. −1 (b) There is an invertible S 2 Mn such that S AS = (bij) is in upper triangular form with jbijj < " for all 1 ≤ i < j ≤ n. Theorem If A and B commute, then there is a unitary U such that U ∗AU and U ∗BU are in upper triangular form. Consequently, one can label the eigenvalues of A and B as a1; : : : ; an and b1; : : : ; bn so that A + B have eigenvalues a1 + b1; : : : ; an + bn. 1 Theorem Two matrices A and B are simultaneously triangularizable if and only if one can label the eigenvalues of A and B as a1; : : : ; an and b1; : : : ; bn so that p(A; B) have eigenvalues p(a1; b1); : : : ; p(an; bn) for any polynomial p(x; y). ∗ ∗ Definition Let A 2 Mn. The matrix A is normal if AA = A A. The matrix A is Hermitian if A = A∗. Theorem Let A 2 Mn have eigenvalues a1; : : : ; an. The following are equivalent. (a) A is normal. (b) A is unitarily diagonalizable. ∗ Pn 2 (c) tr A A = j=1 jajj . (d) A has an orthonormal set of n eigenvectors. Corollary A matrix A 2 Mn is Hermitian if and only if it is unitarily similar to a real diagonal matrix. Theorem Let A 2 Mn(R). (a) The matrix A is symmetric if and only if it is orthogonally similar to a diagonal matrix. (b) The matrix is skew-symmetric if and only if it is orthogonally similar to a direct sum of 0 b matrices of the form and a zero matrix. −b 0 (c) The matrix is orthogonal if and only if it is orthogonally similar to a direct sum of matrices cos t sin t of the form and a diagonal orthogonal matrix. − sin t cos t (d) The matrix A is normal if and only if A is orthogonally similar to a direct sum of matrices a b of the form and a (real) diagonal matrix. −b a Theorem Let A 2 Mn;m with n ≥ m. Then A = QR where Q 2 Mn;m has orthogonal columns and R 2 Mm is upper triangular. If A is real, then Q and R can be chosen to be real. Remarks (a) QR can be done by Householder transforms I − 2vv∗, where v is a unit vector. (b) Every unitary matrix U 2 Mn is a product of at most (n − 1)(n − 2)=2 unitary matrices each of them differs from I2 by a 2 × 2 principal submatrix. 2.