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). Each function works by calling the function OrthogonalizeGram on its argument and then determining whether the resulting diagonal matrix has the appropriate form.

IsPositiveDefinite(F) : ModMatRngElt -> BoolElt

IsPositiveDefinite(F) : AlgMatElt -> BoolElt

Given a symmetric matrix F belonging to the matrix module S=öm_R(M, M) or the matrix algebra S=M_n(R), where R is a subring of the real field, return whether F is positive definite, i.e., whether vFv^(tr) > 0 for all non-zero vectors v in R^n.

IsPositiveSemiDefinite(F) : ModMatRngElt -> BoolElt

IsPositiveSemiDefinite(F) : AlgMatElt -> BoolElt

Given a symmetric matrix F belonging to the matrix module S=öm_R(M, M) or the matrix algebra S=M_n(R), where R is a subring of the real field, return whether F is positive semi-definite, i.e., whether vFv^(tr) >= 0 for all non-zero vectors v in R^n.

IsNegativeDefinite(F) : ModMatRngElt -> BoolElt

IsNegativeDefinite(F) : AlgMatElt -> BoolElt

Given a symmetric matrix F belonging to the matrix module S=öm_R(M, M) or the matrix algebra S=M_n(R), where R is a subring of the real field, return whether F is negative definite, i.e., whether vFv^(tr) < 0 for all non-zero vectors v in R^n.

IsNegativeSemiDefinite(F) : ModMatRngElt -> BoolElt

IsNegativeSemiDefinite(F) : AlgMatElt -> BoolElt

Given a symmetric matrix F belonging to the matrix module S=öm_R(M, M) or the matrix algebra S=M_n(R), where R is a subring of the real field, return whether F is negative semi-definite, i.e., whether vFv^(tr) <= 0 for all non-zero vectors v in R^n.