Homework Help: Symmetric positive definite matrix

1. Aug 25, 2012

IniquiTrance

1. The problem statement, all variables and given/known data
(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 $A=B^2$.

2. Relevant equations

3. 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!

2. Aug 26, 2012

MathematicalPhysicist

Why should it be different?

http://en.wikipedia.org/wiki/Positive-definite_matrix

semi positive means non-negative and positive is positive, this means that the eigenvalues are either way ≥0, so you can define a square root of a diagonal matrix in both cases as the square roots of the eigenvalues.

3. Aug 27, 2012

Thanks.