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## Homework Statement

(i) Show that if [itex]A[/itex] is symmetric positive

**semi-definite**, then there exists a symmetric matrix [itex]B[/itex] such that [itex]A=B^2[/itex].

(ii) Let [itex]A[/itex] be symmetric positive

**definite**. Find a matrix [itex]B[/itex] such that [itex]A=B^2[/itex].

## Homework Equations

## The Attempt at a Solution

For part 1, I used:

[tex] B = Q\sqrt{\Lambda} Q^T[/tex]

So that,

[tex]B^T B = (Q\sqrt{\Lambda} Q^T)^T Q\sqrt{\Lambda} Q^T[/tex]

[tex]=Q\sqrt{\Lambda} Q^T Q\sqrt{\Lambda} Q^T[/tex]

[tex]=Q\Lambda Q^T[/tex]

[tex]=A[/tex]

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