Math 408 Advanced Linear Algebra Chi-Kwong Li

Chapter 2 Unitary similarity and normal matrices n ∗ Definition A set of vector {x1, . . . , xk} 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 {x1, . . . , xn} in C is an orthonormal basis. A matrix U ∈ Mn ∗ is unitary if it has orthonormal columns, i.e., U U = In.

Theorem Let U ∈ 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 ∈ 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 ∈ Mn are unitarily similar if A = U BU for some unitary U.

Theorem If A, B ∈ Mn are unitarily similar, then

X 2 ∗ ∗ X 2 |aij| = tr (A A) = tr (B B) = |bij| . 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 ∈ Mn is unitarily similar to a upper triangular matrix.

Theorem Every A ∈ 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 ∈ Mn and ε > 0.

(a) There is A˜ ∈ Mn with distinct eigenvalues such that kA − A˜kF < ε, where kXkF = (tr X∗X)1/2. −1 (b) There is an invertible S ∈ Mn such that S AS = (bij) is in upper triangular form with

|bij| < ε 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 ∈ Mn. The matrix A is normal if AA = A A. The matrix A is Hermitian if A = A∗.

Theorem Let A ∈ 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 |aj| . (d) A has an orthonormal set of n eigenvectors.

Corollary A matrix A ∈ Mn is Hermitian if and only if it is unitarily similar to a real diagonal matrix.

Theorem Let A ∈ 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 ∈ Mn,m with n ≥ m. Then A = QR where Q ∈ Mn,m has orthogonal columns and R ∈ 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 ∈ 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.

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