The IsDefinite(A, query = 'positive_definite') returns true if $A$ is a real symmetric or a complex Hermitian Matrix and all the eigenvalues are determined to be positive.  This command is equivalent to IsDefinite(A), that is, the default query is for positive definiteness.

Similarly, for real symmetric or complex Hermitian Matrices, the following calling sequences return the indicated result.

IsDefinite(A, query = 'positive_semidefinite') returns true if all the eigenvalues are determined to be non-negative.

IsDefinite(A, query = 'negative_definite') returns true if all the eigenvalues are determined to be negative.

IsDefinite(A, query = 'negative_semidefinite') returns true if all the eigenvalues are determined to be non-positive.

If the eigenvalues are determined to be other than described in the cases above, a value of false is returned.

If any of the conditions on the eigenvalues cannot be resolved, a boolean expression representing the condition which must be satisfied for the query to resolve to "true" is returned.

The definition of  positive definite is that, for all column Vectors $x$, $x^{*}.A.x>0$, where  $x^{*}$ is the Hermitian transpose of $x$.

The definitions for positive semidefinite, negative definite, and negative semidefinite involve reversal of the inequality sign, or relaxation from a strict inequality.

For real non-symmetric (complex non-Hermitian) Matrices, definiteness is established by considering the symmetric (Hermitian) part of $A$, that is,  $\frac{1}{2}\left(A+A^{+}\right)$ ($\frac{1}{2}\left(A+A^{*}\right)$).

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$A≔\mathrm{DiagonalMatrix}\left(\left[-5\,0\,-1\right]\right)$

$A≔\left[\begin{array}{ccc}\mathrm{-5}& 0& 0\\ 0& 0& 0\\ 0& 0& \mathrm{-1}\end{array}\right]$

$\mathrm{IsDefinite}\left(A\right)$

$\mathrm{false}$

$\mathrm{IsDefinite}\left(A\,'\mathrm{query}'='\mathrm{positive\_semidefinite}'\right)$

$\mathrm{false}$

$\mathrm{IsDefinite}\left(A\,'\mathrm{query}'='\mathrm{negative\_semidefinite}'\right)$

$\mathrm{true}$

$B≔⟨⟨1\,8\,3⟩\left|⟨-4\,5\,2⟩\right|⟨6\,1\,0⟩⟩$

$B≔\left[\begin{array}{ccc}1& \mathrm{-4}& 6\\ 8& 5& 1\\ 3& 2& 0\end{array}\right]$

$\mathrm{IsDefinite}\left(B\right)$

$\mathrm{false}$

$C≔⟨⟨1\,2+\mathrm{I}⟩|⟨2-\mathrm{I}\,5⟩⟩$

$C≔\left[\begin{array}{cc}1& 2-\mathrm{I}\\ 2+\mathrm{I}& 5\end{array}\right]$

$\mathrm{IsDefinite}\left(C\right)$

$\mathrm{false}$

$\mathrm{IsDefinite}\left(C\,'\mathrm{query}'='\mathrm{positive\_semidefinite}'\right)$

$\mathrm{true}$

$\mathrm{IsDefinite}\left(C\,'\mathrm{query}'='\mathrm{negative\_semidefinite}'\right)$

$\mathrm{false}$