issymmetric

Determine if matrix is symmetric or skew-symmetric

Syntax

tf = issymmetric(A)

tf = issymmetric(A,skewOption)

Description

tf = issymmetric(A) returns logical 1 (true) if A is a symmetric matrix. Otherwise, it returns logical 0 (false).

tf = issymmetric(A,skewOption) specifies the type of the test. Specify skewOption as "skew" to determine if A is skew-symmetric.

Examples

Test If Hermitian Matrix Is Symmetric

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 Hermitian and has a real-valued diagonal.

Test if the matrix is symmetric.

tf = issymmetric(A)

tf = logical
   0

The matrix A is not symmetric because it is equal to its complex conjugate transpose, A', but not its nonconjugate transpose, A.'.

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

A(3,1) = 1i;

Test if the modified matrix is symmetric.

tf = issymmetric(A)

tf = logical
   1

The matrix A is now symmetric because it is equal to its nonconjugate transpose, A.'.

Test If Matrix Is Skew-Symmetric

Create a 4-by-4 matrix.

A = [0 1 -2 5; -1 0 3 -4; 2 -3 0 6; -5 4 -6 0]

A = 4×4

     0     1    -2     5
    -1     0     3    -4
     2    -3     0     6
    -5     4    -6     0

The matrix is real and has a diagonal of zeros.

Test if the matrix is skew-symmetric by specifying the test type as "skew".

tf = issymmetric(A,"skew")

tf = logical
   1

The matrix A is skew-symmetric because it is equal to the negation of its nonconjugate transpose, -A.'.

Input Arguments

A — Input array

array

Input array. If A is not a square matrix, then issymmetric 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 if A is skew-symmetric.

More About

Symmetric Matrix

• A square matrix, A, is symmetric if it is equal to its nonconjugate transpose, A = A.'.

  In terms of the matrix elements, this means that

  $$a_{i,j}=a_{j,i}.$$

• Since real matrices are unaffected by complex conjugation, a real matrix that is symmetric is also Hermitian. For example, the matrix

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

  is both symmetric and Hermitian.

Skew-Symmetric Matrix

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

  In terms of the matrix elements, this means that

  $$a_{i,j}=-a_{j,i}.$$

• 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-symmetric and skew-Hermitian.

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 Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

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

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

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

The issymmetric function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). 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).

Version History

Introduced in R2014a