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 V to be the span of \{(1,1,1)^t, (1,2,3)^t\} in C^3, and the Hermitian form to be

\langle v, w \rangle=v^t\bar{w}.

We need to see if \langle v,v\rangle >0 for all non-zero v=(v_1,v_2,v_3)^t \in V. But

\langle v, v\rangle =v^t\bar{v}=|v_1|^2+|v_2|^2+|v_3|^2>0

as long as some v_i is non-zero. So the form is positive definite. Notice that this form is actually defined and is positive definite on all of C^3. In general if you have a Hermitian form on a C-vectors space or a symmetric bilinear form on an R-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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