A^2,is a orthogonal matrix,is A a orthogonal matrix?

[muzhiqingfeng]

A^2,is a orthogonal matrix,is A a orthogonal matrix?
If this is not true ,in which case it is true?

 

[galactus]

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

$\displaystyle Ax\cdot Ay=\frac{1}{4}||Ax+Ay||^{2}-\frac{1}{4}||Ax-Ay||^{2}=\frac{1}{4}||A(x+y)||^{2}-\frac{1}{4}A(x-y)||^{2}=\frac{1}{4}||x+y||^{2}-\frac{1}{4}||x-y||^{2}=x\cdot y$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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?.