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I have a following block matrix

M1 =([a11 a12;a21 a22] [b13;b23];[b13;b23]' d33) + ([c -cr]'*[c -cr] 0;0 I)

now what i observe is that whether i use [c -cr]'*[c -cr] or [c cr]'*[c cr] (the sign change in cr the eigen value of M1 doesn't change. How can i prove this in general? Or this is just a co-incidence that, with the examples i am running there is no change in eigen values of M1 with the sign changes. Please I am really looking forward for your answers. Thanks in advance.

All the elements of M1 are matrices of appropriate dimensions.

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The above notations are strange, although I could try to guess what you mean. Can you explain more? Is this some language in Maple, Sage, Mathematica, etc? –  math-visitor Jul 13 '12 at 8:55
    
Thanks for the reply, you can understand it like there is a M1 which is the sum of two matrices e.g., M1 = Ma + Mb. Now for Mb matrix the eigen values doesn't change with changing signs of cr (i can prove this by using a similar matrix technique which P = [I 0 0;0 -I 0;0 0 I] where I is the identity matrix of appropriate dimension... Now the question is that what will be the net effect of no change of eigen value of Mb on M1? I hope it clearfy the question a bit –  Mohsin Jul 13 '12 at 9:04
    
Does $M_1$ look like$$\begin{pmatrix}a_{11}&a_{12}&b_{13}\\a_{21}&a_{22}&b_{23}\\b_{13}&b_{23}&d_{‌​33}\end{pmatrix}$$? –  Host-website-on-iPage Jul 13 '12 at 9:10
    
its like &M1& \begin{align}\begin{pmatrix}a_{11}&0&b_{13}\\0&a_{22}&b_{23}\\b_{13}^\top &b_{23}^\top&d_{33}\end{pmatrix} + \begin{pmatrix}c^\top c &-cc_r&0\\-c_r c&c_rc_r&0\\0 &0 &I\end{pmatrix} \end{align} –  Mohsin Jul 13 '12 at 9:11
    
Also, what's meant by (') followed by a star. I assume (') is transpose? –  Host-website-on-iPage Jul 13 '12 at 9:12

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