fiziks Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics (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 . 34ii 5 T 34i 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 . H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com | Email: [email protected] fiziks Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics 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 31i 2222 22 H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com | Email: [email protected] .