## Square root of a squared block matrix

Hi everybody,

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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 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor Hi GoodSpirit! Have you tried transforming it into the form $$M=\begin{bmatrix} P &0\\ 0 &Q\\ \end{bmatrix}$$
 Hi tiny-tim, Thank you for answering. Thatīs an interesting idea but how do you do that...? It is not easy... I must say that there is more... M is a typical covariance matrix so it is symmetric and semi-positive 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 answer!:) All the best GoodSpirit

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