The IsMatrixShape command verifies whether the matrix A is a certain "shape".

The only types of "shapes" that the IsMatrixShape command can verify are:

Diagonal : shape = diagonal

Strictly diagonally dominant : shape = strictlydiagonallydominant

Diagonally dominant : shape = diagonallydominant

Hermitian : shape = hermitian

Positive definite : shape = positivedefinite

Symmetric : shape = symmetric

Upper or lower triangular : shape = triangular[upper] or shape = triangular[lower], respectively

Tridiagonal : shape = tridiagonal

If neither upper nor lower is specified, the triangular option defaults to triangular[upper].

The Student[NumericalAnalysis] subpackage's definition of positive definiteness is as follows.

A complex n-by-n matrix A is positive definite if and only if A is Hermitian and for all n-dimensional complex vectors v, we have $0<\mathrm{\Re }\left(v^H·A·v\right)$, where $\mathrm{\Re }$ denotes the real part of a complex number.

A real n-by-n matrix A is positive definite if and only if A is symmetric and for all n-dimensional real vectors v, we have $0<v^T·A·v$.

To check another "shape" that is not available with the Student[NumericalAnalysis][IsMatrixShape] command see the general IsMatrixShape command.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right)\&colon;$

$A≔\mathrm{Matrix}\left(\left[\left[2\&comma;-1\&comma;0\&comma;0\right]\&comma;\left[-1\&comma;2\&comma;-1\&comma;0\right]\&comma;\left[0\&comma;-1\&comma;2\&comma;-1\right]\&comma;\left[0\&comma;0\&comma;-1\&comma;2\right]\right]\right)$

$A≔\left[\begin{array}{cccc}2& \mathrm{-1}& 0& 0\\ \mathrm{-1}& 2& \mathrm{-1}& 0\\ 0& \mathrm{-1}& 2& \mathrm{-1}\\ 0& 0& \mathrm{-1}& 2\end{array}\right]$

$B≔\mathrm{Matrix}\left(\left[\left[-1\&comma;0\&comma;0\&comma;0\right]\&comma;\left[-1\&comma;2\&comma;0\&comma;0\right]\&comma;\left[1\&comma;-1\&comma;-3\&comma;0\right]\&comma;\left[-1\&comma;1\&comma;-1\&comma;4\right]\right]\right)$

$B≔\left[\begin{array}{cccc}\mathrm{-1}& 0& 0& 0\\ \mathrm{-1}& 2& 0& 0\\ 1& \mathrm{-1}& \mathrm{-3}& 0\\ \mathrm{-1}& 1& \mathrm{-1}& 4\end{array}\right]$

$C≔\mathrm{Matrix}\left(\left[\left[3\&comma;-\mathrm{I}\&comma;1\&comma;0\right]\&comma;\left[\mathrm{I}\&comma;4\&comma;2\mathrm{I}\&comma;0\right]\&comma;\left[1\&comma;-2\mathrm{I}\&comma;5\&comma;1\right]\&comma;\left[0\&comma;0\&comma;1\&comma;4\right]\right]\right)$

$C≔\left[\begin{array}{cccc}3& \mathrm{-I}& 1& 0\\ \mathrm{I}& 4& 2\mathrm{I}& 0\\ 1& -2\mathrm{I}& 5& 1\\ 0& 0& 1& 4\end{array}\right]$

$\mathrm{IsMatrixShape}\left(A\&comma;'\mathrm{diagonal}'\right)$

$\mathrm{false}$

$\mathrm{IsMatrixShape}\left(A\&comma;'\mathrm{strictlydiagonallydominant}'\right)$

$\mathrm{false}$

$\mathrm{IsMatrixShape}\left(A\&comma;'\mathrm{diagonallydominant}'\right)$

$\mathrm{true}$

$\mathrm{IsMatrixShape}\left(C\&comma;'\mathrm{hermitian}'\right)$

$\mathrm{true}$

$\mathrm{IsMatrixShape}\left(A\&comma;'\mathrm{positivedefinite}'\right)$

$\mathrm{true}$

$\mathrm{IsMatrixShape}\left(B\&comma;'\mathrm{positivedefinite}'\right)$

$\mathrm{false}$

$\mathrm{IsMatrixShape}\left(C\&comma;'\mathrm{positivedefinite}'\right)$

$\mathrm{true}$

$\mathrm{IsMatrixShape}\left(A\&comma;'\mathrm{symmetric}'\right)$

$\mathrm{true}$

$\mathrm{IsMatrixShape}\left(B\&comma;'\mathrm{triangular}'\left['\mathrm{upper}'\right]\right)$

$\mathrm{false}$

$\mathrm{IsMatrixShape}\left(B\&comma;'\mathrm{triangular}'\right)$

$\mathrm{false}$

$\mathrm{IsMatrixShape}\left(\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(B\right)\&comma;'\mathrm{triangular}'\right)$

$\mathrm{true}$

$\mathrm{IsMatrixShape}\left(B\&comma;'\mathrm{triangular}'\left['\mathrm{lower}'\right]\right)$

$\mathrm{true}$

$\mathrm{IsMatrixShape}\left(A\&comma;'\mathrm{tridiagonal}'\right)$

$\mathrm{true}$