Positive definite operator/matrix question

[holezch]

Homework Statement

Prove that T is positive definite if and only if

\sum_{i,j} A_{ij}a_{j}\bar{a_{i}} > 0
for any non-zero tuple (a1, ...... , an )

Let A be [ T ]_{\beta}

where \beta is an orthogonal basis for T

The Attempt at a Solution

the sum looked like the matrix multiplication of a n-tuple and a matrix A, so I looked into that and couldn't get anything.. any hints please? I'm also struggle to realize what significant that sum could have, right now it doesn't even mean anything to me.

Thanks!

 

[Office_Shredder]

If the capital A's are supposed to be terms in matrix, you might want to think about

\overline{a}^tAa

 

[holezch]

If the capital A's are supposed to be terms in matrix, you might want to think about

\overline{a}^tAa

thanks, someone else told me to think about a^{*} A a , but I'm not sure why I'd be considering this. I basically have these things to work with: that if T is positive definite ( one way of the implication ), then all its eigenvalues are positive,T is self-adjoint and < Tx , x > > 0.

Thanks

 

[holezch]

I'm also confused a*A might not make sense.. if a is a column vector in F^n and A is a matrix?