1. The problem statement, all variables and given/known data Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ 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 R3. 2. Relevant equations None 3. The attempt at a solution I've shown that x · Dy is an inner product. I know that Z-1 is equal to ZT. I believe that will lead me somewhere. I'm just having trouble showing the property <x, x> ≥ 0. I also know that (ATx)⋅x = x ⋅ Ax. Just missing one step. I don't know what it is.