Positive definite inner products

Hi Sir,

I’ve got another question.

When determining whether an inner product (in the Real space) is positive definite, we check it’s Real Canonical Form. If it’s diagonals are all “1”s, then it is positive definite. What about determining positive definiteness in the Complex space? I realised(from Sheet 8, Qns 1e) that the same method wouldn’t work (find the complex canonical form and see if its diagonal entries are all “1”s). How can we determine if an inner product is positive definite in the complex space?

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Reply:

Your observation is correct. Note that in the complex space, the positive definiteness is being checked for a *Hermitian* form, not a symmetric bilinear form (what is the difference?). Hence, the whole discussion of complex canonical forms does not apply. The only answer I can give at the moment is that you should think about the definition carefully. Of course there are the problems from the coursework, but it may help already to ask yourself: Is the standard Hermitian form on C^n,

\langle v,w \rangle := v_1\bar{w}_1+v_2\bar{w}_2+\cdots v_n\bar{w}_n

positive definite? Why or why not?

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