Proof that a matrix can be written as the product of a positive definite matrix and an orthogonal matrix How can I show that for a real invertible matrix $A$ with dimensions $n \times  n$
$A$ can be written:
$A = B C$
where $B$ is a positive definite matrix and $C$ is an orthogonal matrix.

This is called the (left) polar decomposition of $A$. Note that if $A = BC$ where $B$ is positive definite and $C$ is orthogonal then
$$ A^T A = (BC)^T (BC) = C^T B^T B C = C^T C = C^2. $$
This means that you have $C = \sqrt{A^T A}$ (where $\sqrt{A^T A}$ is the unique positive definite root of $A^T A$) and $B = AC^{-1} = A(\sqrt{A^T A})^{-1}$. Going backwards, in order to show that such a decomposition exist define
$$ C := \sqrt{A^T A}, \,\,\, B := A(\sqrt{A^T A})^{-1} $$
and show that this is indeed well-defined, that $B$ is orthogonal and that $C$ is positive definite and that $A = BC$.
