# Symmetric positive definite matrix

## Homework Statement

(i) Show that if $A$ is symmetric positive semi-definite, then there exists a symmetric matrix $B$ such that $A=B^2$.

(ii) Let $A$ be symmetric positive definite. Find a matrix $B$ such that $A=B^2$.

## The Attempt at a Solution

For part 1, I used:

$$B = Q\sqrt{\Lambda} Q^T$$

So that,

$$B^T B = (Q\sqrt{\Lambda} Q^T)^T Q\sqrt{\Lambda} Q^T$$
$$=Q\sqrt{\Lambda} Q^T Q\sqrt{\Lambda} Q^T$$
$$=Q\Lambda Q^T$$
$$=A$$

I am assuming this cannot be used for part 2. Any help is much appreciated!