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ishermitian

Determine if matrix is Hermitian or skew-Hermitian

Syntax

Description

example

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

example

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

Examples

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Create a 3-by-3 matrix.

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

   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'.

Create a 3-by-3 matrix.

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

   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

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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

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

More About

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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

    ai,j=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=[100210101]

    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

    ai,j=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=[0110]

    is both skew-Hermitian and skew-symmetric.

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

Introduced in R2014a

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