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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) : AlgMatElt -> BoolElt
Given a symmetric matrix F belonging to the
matrix module S=HomR(M, M) or the matrix algebra S=Mn(R),
where R is a subring of the real field,
return whether F is positive definite,
i.e., whether vFvtr > 0 for all non-zero vectors v∈Rn.
IsPositiveSemiDefinite(F) : AlgMatElt -> BoolElt
Given a symmetric matrix F belonging to the
matrix module S=HomR(M, M) or the matrix algebra S=Mn(R),
where R is a subring of the real field,
return whether F is positive semi-definite,
i.e., whether vFvtr ≥0 for all non-zero vectors v∈Rn.
IsNegativeDefinite(F) : AlgMatElt -> BoolElt
Given a symmetric matrix F belonging to the
matrix module S=HomR(M, M) or the matrix algebra S=Mn(R),
where R is a subring of the real field,
return whether F is negative definite,
i.e., whether vFvtr < 0 for all non-zero vectors v∈Rn.
IsNegativeSemiDefinite(F) : AlgMatElt -> BoolElt
Given a symmetric matrix F belonging to the
matrix module S=HomR(M, M) or the matrix algebra S=Mn(R),
where R is a subring of the real field,
return whether F is negative semi-definite,
i.e., whether vFvtr ≤0 for all non-zero vectors v∈Rn.
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