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Square root of a squared block matrix 
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#1
Feb713, 12:35 PM

P: 19

Hi everybody,
Im trying to compute the square root of the following squared block matrix: [tex] \begin{equation} M=\begin{bmatrix} A &B\\ C &D\\ \end{bmatrix} \end{equation} [/tex] (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 


#2
Feb713, 01:31 PM

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P: 26,148

Hi GoodSpirit!
Have you tried transforming it into the form [tex] \begin{equation} M=\begin{bmatrix} P &0\\ 0 &Q\\ \end{bmatrix} \end{equation} [/tex] 


#3
Feb913, 06:15 AM

P: 19

Hi tinytim,
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 semipositive definite. A and D are symmetric and positive semidefinite (covariance matrices too) and [tex]B=C^T[/tex] 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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