# Square root of a squared block matrix

I’m trying to compute the square root of the following squared block matrix:

$$M=\begin{bmatrix} A &B\\ C &D\\ \end{bmatrix}$$

(that is $M^{1/2}$) as function of $A$, $B$, $C$, $D$ wich are all square matrices.

Can you help me?

I sincerely thank you! :)

All the best

GoodSpirit

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"The square root" is meaningless - there are potentially infinitely many square roots to a matrix. – Thomas Andrews Feb 7 '13 at 18:33
If $M$ was symmetric and positive definite, then there is a unique positive definite square-root of $M$. For that you need to assume $B=C^\top$ and $A,D$ must be positive definite. – Peder Feb 8 '13 at 6:01
I really thank your answers! Peter, that is precisely the case. M is a covariance matrix but positive semi-definite. A and D are symmetric and positive semi-definite(covariance matrices too) and $$B=C^T$$ and B is the cross covariance matrix of A and D. My attempt is based on eigendecomposition $$M=Q \Lambda Q^T$$ and $$M=\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$ But it lead to something very complicated. I really thank you all for your answers!:) All the best GoodSpirit – GoodSpirit Feb 9 '13 at 11:58