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Square root of a squared block matrix

  1. Feb 7, 2013 #1
    Hi everybody,

    I’m 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. jcsd
  3. Feb 7, 2013 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi GoodSpirit! :smile:

    Have you tried transforming it into the form
    [tex]
    \begin{equation}
    M=\begin{bmatrix}
    P &0\\
    0 &Q\\
    \end{bmatrix}
    \end{equation}
    [/tex]
     
  4. Feb 9, 2013 #3
    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 [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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