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.

If $A=USV^T$ is an SVD of $A$, then $A=(USU^T)(UV^T)=:BC$. Since $A$ is invertible, $S$ is diagonal invertible as well with positive diagonal entries, hence $B=USU^T$ is positive definite. The matrix $C=UV^T$ is a product of two orthogonal matrices and so is orthogonal.
