let V be the vector space of m x n matrices over R. prove that :

phi(A,B) = trace((transpose(B))*A)

defines an inner product in V.

cant work out where to start here. any pointers in the right direction would be appreciated.

thanks

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- February 24th 2007, 06:57 AMmesterpavector space inner product proof
let V be the vector space of m x n matrices over R. prove that :

phi(A,B) = trace((transpose(B))*A)

defines an inner product in V.

cant work out where to start here. any pointers in the right direction would be appreciated.

thanks - February 24th 2007, 09:01 AMPlato
This problem is just an exercise in notation usage. Without TeX it is next to impossible to display. However, here is a bit of guidance.

This idea is to show that this inner product is equivalent to the usual inner product defined on the space nxn matrices. I will give a good reference__Linear Algebra__by Larry Smith. He shows how to work this out for <A,B>=tr{AB^T}. But his work is easily modified to this problem.