Hermitian and Skew - Hermitian Matrices

Hermitian Matrix:

A square matrix A is Hermitian if it is equal to its conjugate transpose: A=A

  • Properties:
    • For a real matrix, being Hermitian is equivalent to being symmetric.
    • For a complex matrix, the elements satisfy aij=aji for all i and j.
    • The diagonal elements must be real.
    • Hermitian matrices are always square matrices.

Example of Hermitian Matrix:

A=[25i5i3]

In this case, A=A, which makes it a Hermitian matrix.

Skew-Hermitian Matrix:

A square matrix A is skew-Hermitian if it is equal to the negative of its conjugate transpose: A=A

  • Properties:
    • For a real matrix, a skew-Hermitian matrix is equivalent to a skew-symmetric matrix.
    • For a complex matrix, the elements satisfy aij=aji for all i and j.
    • The diagonal elements must be purely imaginary.

Example of Skew-Hermitian Matrix:

A=[07i7i0]

In this case, A=A, which makes it a skew-Hermitian matrix.

Properties of Hermitian and Skew-Hermitian Matrices:

  1. Sum of Hermitian Matrices: The sum of two Hermitian matrices is also Hermitian. A+B=C where A, B, and C are Hermitian matrices.

  2. Sum of Skew-Hermitian Matrices: The sum of two skew-Hermitian matrices is also skew-Hermitian. A+B=C where A, B, and C are skew-Hermitian matrices.

  3. Product with Scalar: Both Hermitian and skew-Hermitian matrices preserve their property when multiplied by a scalar.