Chapter 8.2, Problem 28E

Show that any positive definite $n\times n$matrix A can be written as $A=B{B}^T$ , where B is an $n\times n$matrix with orthogonal columns. Hint: There exists an orthogonal matrix S such that $S^{-1}AS=S^{T}AS=D$ is a diagonal matrix with positive diagonal entries. Then $A=SD{S}^T$. Now write D as the square of a diagonal matrix.