ishermitian

Determine if matrix is Hermitian or skew-Hermitian

Syntax

tf = ishermitian(A)

tf = ishermitian(A,skewOption)

Description

tf = ishermitian(A) returns logical 1 (true) if square matrix A is Hermitian; otherwise, it returns logical 0 (false).

tf = ishermitian(A,skewOption) specifies the type of the test. Specify skewOption as 'skew' to determine if A is skew-Hermitian.

Examples

Test if Symmetric Matrix Is Hermitian

Create a 3-by-3 matrix.

A = [1 0 1i; 0 1 0; 1i 0 1]

A = 3×3 complex

   1.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 1.0000i
   0.0000 + 0.0000i   1.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 1.0000i   0.0000 + 0.0000i   1.0000 + 0.0000i

The matrix is symmetric with respect to its real-valued diagonal.

Test whether the matrix is Hermitian.

tf = ishermitian(A)

tf = logical
   0

The result is logical 0 (false) because A is not Hermitian. In this case, A is equal to its transpose, A.', but not its complex conjugate transpose, A'.

Change the element in A(3,1) to be -1i.

A(3,1) = -1i;

Determine if the modified matrix is Hermitian.

tf = ishermitian(A)

tf = logical
   1

The matrix, A, is now Hermitian because it is equal to its complex conjugate transpose, A'.

Test if Matrix Is Skew-Hermitian

Create a 3-by-3 matrix.

A = [-1i -1 1-i;1 -1i -1;-1-i 1 -1i]

A = 3×3 complex

   0.0000 - 1.0000i  -1.0000 + 0.0000i   1.0000 - 1.0000i
   1.0000 + 0.0000i   0.0000 - 1.0000i  -1.0000 + 0.0000i
  -1.0000 - 1.0000i   1.0000 + 0.0000i   0.0000 - 1.0000i

The matrix has pure imaginary numbers on the main diagonal.

Specify skewOption as 'skew' to determine whether the matrix is skew-Hermitian.

tf = ishermitian(A,'skew')

tf = logical
   1

The matrix, A, is skew-Hermitian since it is equal to the negation of its complex conjugate transpose, -A'.

Input Arguments

A — Input matrix

numeric matrix

Input matrix, specified as a numeric matrix. If A is not square, then ishermitian returns logical 0 (false).

Data Types: single | double | logical
Complex Number Support: Yes

skewOption — Test type

'nonskew' (default) | 'skew'

Test type, specified as 'nonskew' or 'skew'. Specify 'skew' to test whether A is skew-Hermitian.

More About

Hermitian Matrix

• A square matrix, A, is Hermitian if it is equal to its complex conjugate transpose, A = A'.

  In terms of the matrix elements, this means that

  $$a_{i,j}={\overline{a}}_{j,i}.$$

• The entries on the diagonal of a Hermitian matrix are always real. Since real matrices are unaffected by complex conjugation, a real matrix that is symmetric is also Hermitian. For example, the matrix

  $$A=\left[\begin{array}{cc}\begin{array}{c}1\\ 0\end{array}& \begin{array}{cc}\begin{array}{c}0\\ 2\end{array}& \begin{array}{c}1\\ 0\end{array}\end{array}\\ 1& \begin{array}{cc}0& 1\end{array}\end{array}\right]$$

  is both symmetric and Hermitian.

• The eigenvalues of a Hermitian matrix are real.

Skew-Hermitian Matrix

• A square matrix, A, is skew-Hermitian if it is equal to the negation of its complex conjugate transpose, A = -A'.

  In terms of the matrix elements, this means that

  $$a_{i,j}=-{\overline{a}}_{j,i}.$$

• The entries on the diagonal of a skew-Hermitian matrix are always pure imaginary or zero. Since real matrices are unaffected by complex conjugation, a real matrix that is skew-symmetric is also skew-Hermitian. For example, the matrix

  $$A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$$

  is both skew-Hermitian and skew-symmetric.

• The eigenvalues of a skew-Hermitian matrix are purely imaginary or zero.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Code generation does not support sparse matrix inputs for this function.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).