The product of orthogonal matrices is orthogonal
They preserve orthogonality and thus form a group.
So, if A is orthogonal, then A^2 is also orthogonal.
Assume that \(\displaystyle ||Ax||=||x||\) for all x in R^n. So, we have
\(\displaystyle Ax\cdot Ay=\frac{1}{4}||Ax+Ay||^{2}-\frac{1}{4}||Ax-Ay||^{2}=\frac{1}{4}||A(x+y)||^{2}-\frac{1}{4}A(x-y)||^{2}=\frac{1}{4}||x+y||^{2}-\frac{1}{4}||x-y||^{2}=x\cdot y\)
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Let's say \(\displaystyle R^{m}\rightarrow W\) be the orthogonal projection of \(\displaystyle R^{m}\) onto a subspace W.
Let's say we have \(\displaystyle [P]=A(A^{T}A)^{-1}A^{T}\)
where A is any matrix formed using a set of basis vectors for W as its column vectors.
Therefore, \(\displaystyle [P]^{2}=[A(A^{T}A)^{-1}A^{T}][A(A^{T}A)^{-1}A^{T}]\)
\(\displaystyle =A[(A^{T}A)^{-1}(A^{T}A)](A^{T}A)^{-1}A^{T}\)
\(\displaystyle =A(A^{T}A)^{-1}A^{T}\)
\(\displaystyle =[P]\)
Does that help?.
