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., Z, Q, or a real field), though the SemiDefinite functions must be over Z or 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∈Rⁿ. 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 ≥0 for all non-zero vectors v∈Rⁿ.

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∈Rⁿ. 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 ≤0 for all non-zero vectors v∈Rⁿ.

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.