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).

example

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

example

Examples

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Test If Hermitian Matrix Is Symmetric

Open Live Script

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

Open Live Script

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

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`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

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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
$A = [\begin{matrix} \begin{matrix} \begin{matrix} 1 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 2 \end{matrix} \end{matrix} & \begin{matrix} 1 \\ 0 \end{matrix} \\ \begin{matrix} 1 & 0 \end{matrix} & 1 \end{matrix}]$
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
$A = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]$
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

issymmetric

Syntax

Description

Examples

Test If Hermitian Matrix Is Symmetric

Test If Matrix Is Skew-Symmetric

Input Arguments

A — Input array array

skewOption — Test type "nonskew" (default) | "skew"

More About

Symmetric Matrix

Skew-Symmetric Matrix

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

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

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

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

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

Version History

See Also

`A` — Input array
array

`skewOption` — Test type
`"nonskew"` (default) | `"skew"`

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

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

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.

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

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