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Math Help - Non-definite matrices? HELP!

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    Non-definite matrices? HELP!

    So I am trying to determine if the difference of two matrices is positive(negative) semi-definite. The matrices, we will call them W and \Omega, are partitioned as

    W=\begin{pmatrix}W_{11}&W_{12}\\W_{21}&W_{22}\end{  pmatrix}
    and
    \Omega=\begin{pmatrix}W_{11}&W_{12}-B\\W_{21}-B&W_{22}+B-B\cdot{W_{21}}-{W_{21}}\cdot{B}\end{pmatrix}..

    My problem is that I wish to determine the definiteness of the matrix A_{1}=\Omega-W or/and A_{2}=W-\Omega. These matrices are given by
    A_{1}=\begin{pmatrix}0&-B\\-B&B-B\cdot{W_{21}}-{W_{21}}\cdot{B}\end{pmatrix}
    and
    A_{1}=\begin{pmatrix}0&B\\B&-B+B\cdot{W_{21}}+{W_{21}}\cdot{B}\end{pmatrix}.


    We know from simple linear algebra that a matrix is positive(negative) semi-definite if and only if all of its principle minors are non-negative(non-positive). With this I have two questions: 1) Does the same hold true for block matrices? 2) If this does hold true the second principle minor, in block form, for both matrices is -BB, what does this mean exactly. Does this mean that the matrices are non-definite?



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    Re: Non-definite matrices? HELP!

    Ok, I found out that I can use the principle minor characterization of block matrices. However, there is still a problem. While the determinant of the first principle minor is zero. The determinant of the entire matrix(in block form) is det(A_{1})=det(-B\cdot{B}).

    If B where square we have det(A_{1})=det(-B)\cdot{det(B)}. However, in general B is not a square matrix.

    Does anyone know another way for me to break up det(-B\cdot{B})?
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  3. #3
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    Re: Non-definite matrices? HELP!

    Quote Originally Posted by frazdt View Post
    So I am trying to determine if the difference of two matrices is positive(negative) semi-definite. The matrices, we will call them W and \Omega, are partitioned as

    W=\begin{pmatrix}W_{11}&W_{12}\\W_{21}&W_{22}\end{  pmatrix}
    and
    \Omega=\begin{pmatrix}W_{11}&W_{12}-B\\W_{21}-B&W_{22}+B-B\cdot{W_{21}}-{W_{21}}\cdot{B}\end{pmatrix}.
    Something strange is going on here. For the (1,2)-element W_{12}-B of \Omega to be defined, the matrices W_{12} and B must be the same size (same number of rows and columns). Looking at the other elements of \Omega, you see that B,\,W_{21},\,W_{22},\,BW_{21} and W_{21}B must all be the same size. That implies that they are all square matrices of the same size.

    In any case, the matrix -BB is only defined if B is a square matrix. In general, there is nothing you can say about the positivity or otherwise of -B^2. It could be positive definite, negative definite or (more probably) neither.
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