# 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!

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MathematicalPhysicist
Gold Member
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.

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.
Thanks.