## Positive-definiteness

Dear Dr. Kim,

Could you show me how to determine whether it is positive definite for homework 8 question 1c please.

It will be very grateful if you can show me in your course blog.

Many thanks for your help.

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Reply:

The question takes the vector space to be the span of in , and the Hermitian form to be

We need to see if for all non-zero . But

as long as some is non-zero. So the form is positive definite. Notice that this form is actually defined and is positive definite on all of . In general if you have a Hermitian form on a -vectors space or a symmetric bilinear form on an -vector space, if the original form is positive definite, then its restriction to any subspace is also positive definite, simply by exmaining the definition of positive-definiteness.

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