Orthogonal Matrix Preserves Dot Product: Proof and Explanation

We know that for any two vectors 'u' and 'v' in R^n, their dot product is given by:

<u, v> = u^T v

where (^T) denotes the transpose of a matrix.

Now, let's consider the left-hand side of the given equation:

<Aα, Aβ> = (Aα)^T (Aβ)

= α^T A^T Aβ

since 'A' is an orthogonal matrix, we have A^T A = I, the identity matrix. Therefore,

<Aα, Aβ> = α^T I β = α^T β

which is exactly the right-hand side of the given equation.

Hence, we have shown that <Aα, Aβ> = <α, β> holds for any n*n orthogonal matrix 'A'.