For positive definiteness, take a nondegenerate representation of .

(This identification turns the positive semi-definiteness above into positive definiteness.)

This is not to be confused with a seminorm or pseudonorm, where the norm axioms are satisfied except for positive definiteness.

Conditions 1 and 2 together produce positive definiteness.

However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness.

For Hermitian matrices, the leading principal minors can be used to test for positive definiteness.

This is then a metric; the positive definiteness follows of the injectivity of the differential of an immersion.

The first two conditions listed above on the signs of these minors are the conditions for the positive or negative definiteness of the Hessian.

Note that condition 1 and 2 together produce positive definiteness

Hence, can be compared to zero, and the requirement of positive definiteness in the definition of inner product makes sense.