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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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    Now the problem is with the signs of 'c^\top c_r' and 'c_r^\top c' as they change their signs the eigen values doesn't change. what will be the effect of this on the whole M1... this is my question. I hope that i am able to put my question correctly. I feel extremely sorry for this mess of comments :(2012-07-13

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