For A∈GLn(R), AT=A, is yTAz an inner product?

manpreet Best Answer 2 years ago

I'm reading in my Linear Algebra syllabus, and they write that given A∈GLn(R)A∈GLn(R) and AT=AAT=A we can define the following inner product: (y,z)A=yTAz(y,z)A=yTAz for y,z∈Rny,z∈Rn.

I am a bit in doubt about this. Surely this map is linear in the first component and symmetric. However, I am in doubt about the map being positive-definite. Using the spectral theorem for symmetric matrices, we can find an orthonormal basis {v1,…,vn}{v1,…,vn}, and writing x=∑ni=1civix=∑i=1ncivi and writing out (x,x)A(x,x)A yields ∑nn id="MathJax-Span-123" class="mi" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: normal; font-family: MathJax_Math-italic; font-size: 11.7715px; vertical-align: 0px; box-s

manpreet 2 years ago