# Square root of a squared block matrix

by GoodSpirit
Tags: block, matrix, root, square, squared
 P: 19 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
 Sci Advisor HW Helper Thanks P: 26,148 Hi GoodSpirit! Have you tried transforming it into the form $$M=\begin{bmatrix} P &0\\ 0 &Q\\ \end{bmatrix}$$
 P: 19 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