1. ## 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 $\displaystyle W$ and $\displaystyle \Omega$, are partitioned as

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

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

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 $\displaystyle -BB$, what does this mean exactly. Does this mean that the matrices are non-definite?

Thanks

2. ## 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 $\displaystyle det(A_{1})=det(-B\cdot{B})$.

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

Does anyone know another way for me to break up $\displaystyle det(-B\cdot{B})$?

3. ## Re: Non-definite matrices? HELP!

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

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

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