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## Homework Statement

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Let Z be any 3×3 orthogonal matrix and let A = Z

^{-1}DZ where D is a diagonal matrix with positive integers along its diagonal.

Show that the product <x, y>

_{A}= x · Ay is an inner product for R

^{3}.

## Homework Equations

None

## The Attempt at a Solution

I've shown that x · Dy is an inner product. I know that Z

^{-1}is equal to Z

^{T}. I believe that will lead me somewhere. I'm just having trouble showing the property <x, x> ≥ 0. I also know that (A

^{T}x)⋅x = x ⋅ Ax.

Just missing one step. I don't know what it is.

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