Square root of a squared block matrix

[GoodSpirit]

Hi everybody,

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

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

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

Can you help me?

I sincerely thank you! :)

All the best

GoodSpirit

 

[tiny-tim]

Hi GoodSpirit!

Have you tried transforming it into the form
$$\begin{equation} M=\begin{bmatrix} P &0\\ 0 &Q\\ \end{bmatrix} \end{equation}$$

 

[GoodSpirit]

Hi tiny-tim,

Thank you for answering.
That´s 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