Symmetric positive definite matrix

[IniquiTrance]

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.

Homework Equations

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!

 

[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.

 

[IniquiTrance]

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.