Testing Matrices for Definiteness

The functions in this section test matrices for positive definiteness, etc. They may applied to any symmetric matrix over a real subring (i.e., \({\mathbb{Z}}\), \({\mathbb{Q}}\), or a real field), though the SemiDefinite functions must be over \({\mathbb{Z}}\) or \({\mathbb{Q}}\). Each function works by calling the function OrthogonalizeGram on its argument and then determining whether the resulting diagonal matrix has the appropriate form. Over a real field, a numerical check is made that the matrix has enough stability to determine definiteness.

IsPositiveDefinite(F): Mtrx -> BoolElt

Given a symmetric matrix \(F\) over the rationals or integers or a real field, return whether \(F\) is positive definite, i.e., whether \(vFv^{tr} > 0\) for all non-zero vectors \(v\in {\mathbb{R}}^n\). Over a real field, it will fail in numerically unstable situations.

IsPositiveSemiDefinite(F): Mtrx -> BoolElt

Given a symmetric matrix \(F\) over the rationals or integers, return whether \(F\) is positive semi-definite, i.e., whether \(vFv^{tr} \geq 0\) for all non-zero vectors \(v\in {\mathbb{R}}^n\).

IsNegativeDefinite(F): Mtrx -> BoolElt

Given a symmetric matrix \(F\) over the rationals or integers or a real field, return whether \(F\) is negative definite, i.e., whether \(vFv^{tr} < 0\) for all non-zero vectors \(v\in {\mathbb{R}}^n\). Over a real field, it will fail in numerically unstable situations.

IsNegativeSemiDefinite(F): Mtrx -> BoolElt

Given a symmetric matrix \(F\) over the rationals or integers, return whether \(F\) is negative semi-definite, i.e., whether \(vFv^{tr} \leq 0\) for all non-zero vectors \(v\in {\mathbb{R}}^n\).

Signature(F): Mtrx -> RngIntElt, RngIntElt, RngIntElt

Given a symmetric matrix \(F\) over the rationals or integers, return the signature of \(F\), i.e. the number of positive, negative and zero eigenvalues.

NumericalSignature(M): Mtrx -> RngIntElt, RngIntElt

Given a symmetric matrix over a real field, return its signature, that is, the number of positive and negative eigenvalues. Fails if (at least) one of the eigenvalues is too close to zero.