\(\text{IsHermitian}\)

You can use the \(\text{IsHermitian}\) to determine if a matrix is a Hermitian matrix.

You can use the \ishermitian backslash command to insert this function.

The following variants of this function are available:

  • \(\text{boolean } \text{IsHermitian} \left ( \text{<matrix>} \right )\)

  • \(\text{boolean } \text{IsHermitian} \left ( \text{<matrix>}, \text{<relative tolerance>} \right )\)

The \(\text{<matrix>}\) parameter is the matrix to be tested. The optional \(\text{<relative tolerance>}\) parameter specifies the tolerance to apply when comparing values. If not specified, the \(\text{IsHermitian}\) function will use a tolerance of \(1.0 \times 10^{-8}\).

A matrix, \(M\), is considered Hermitian if the matrix is square and \(M _ { i, j } = \left ( M _ { j, i } \right ) ^ {*}\). This function relaxes this constraint by allowing a test within a given tolerance. The \(\text{IsHermitian}\) function considers a matrix to be Hermitian if the matrix is square and:

\[\left \vert M _ { i,j } - \left ( M _ { j,i } \right ) ^ {*} \right \vert \leq \frac{1}{2} \left \vert M _ { i,j } + \left ( M _ { j,i } \right ) ^ {*} \right \vert t _ {rel}\]

For all \(1 \leq i \leq \text{nrows} \left ( M \right )\) and \(1 \leq j \leq i\) where \(t _ {rel}\) is the relative tolerance. Note that this test is only applied if \(M _ { i,j } \ne \left ( M _ { j,i } \right ) ^ {*}\).

Figure 157 shows the basic use of the \(\text{IsHermitian}\) function.

../_images/is_hermitian_example.png

Figure 157 Example Use Of The IsHermitian Function