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(g) Hermitian, Skew-Hermitian, and Unitary Matrices (Complex Matrices) The complex conjugate of a matrix A is formed by taking the complex conjugate of each element. Thus, Aa ij .

T For the conjugate transpose, we use the notation A  aij .

34ii 5 T 34i 7 Example: A  , then A    762i  562ii  Hermitian Matrix  A square matrix A  aij is called Hermitian if T AA , that is aajiij

 If A is Hermitian, the entries on the main diagonal must satisfy aajjjj , that is they are real.  If a Hermitian matrix is real, then AAATT  . Hence a real Hermitian matrix is a symmetric matrix.  The eigenvalues of a Hermitian matrix (and thus a symmetric matrix) are real. 413 i Example: A  . The eigenvalues are 9, 2 . 13 i 7 Skew- Hermitian Matrix  A square matrix A  aij is called skew-Hermitian if T AA  , that is aajiij 

 If A is skew-Hermitian, then entries on the main diagonal must satisfy aajjjj  , hence

a jj must be pure imaginary or 0.

 If a skew-Hermitian matrix is real, then AATT  A. Hence a real skew-Hermitian matrix is a skew-symmetric matrix.  The eigenvalues of a skew-Hermitian matrix (and thus a skew-symmetric matrix) are pure imaginary or 0 . 32ii Example: A  . 2 ii  The eigenvalues are 4,ii 2 .

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Unitary Matrix  A square matrix A  aij is called unitary if AAT  1  If a unitary matrix is real, then AAATT  1 . Hence a real unitary matrix is an orthogonal matrix.  The eigenvalues of a unitary matrix (and thus an orthogonal matrix) have absolute value 1. 11 i 3 22 Example: A  . 11 3 i 22 1111 11 The eigenvalues are 3,ii 3  and  31i 2222 22

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