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Let $M\in \mathbb{R}^{n\times n}$. And $$B=\begin{bmatrix} -\theta M-M+c_1 I^{n\times n}& \theta M-c_2I^{n\times n}\\ I^{n\times n}& 0^{n\times n} \end{bmatrix};\quad c_1, c_2,\theta \in \mathbb{R}$$

Ben Grossmann
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Vivian
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  • Note that, in general, $$|A|2 = \lambda{max}(\sqrt{A'A}) = \sqrt{\lambda_{max}(A'A)} \neq \lambda_{max}(A)$$ Also, how are you supposed to evaluate $M+c_1$ if $M$ is a matrix and $c_1$ is a scalar? – Ben Grossmann Dec 17 '13 at 13:13
  • @Omnomnomnom, thank you for your advice. – Vivian Dec 17 '13 at 15:59
  • Are we given any information about $M$? For example, is $M$ known to be normal or symmetric? It seems that my approach is not generally particularly helpful – Ben Grossmann Dec 19 '13 at 14:50

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Assuming we have $$ B = \pmatrix{ -(1+\theta)M + c_1I & \theta M - c_2 I\\ I & 0 } $$ It follows that $$ B'B=\pmatrix{ (-(1+\theta)M + c_1I)'(-(1+\theta)M + c_1I)+I & (-(1+\theta)M + c_1I)'(\theta M - c_2I)\\ (-(1+\theta)M + c_1I)(\theta M - c_2I)' & (\theta M - c_2I)(\theta M - c_2I)' } $$ Perhaps expanding the entries will give you some insight into the nature of the eigenvalues of $B'B$.

Ben Grossmann
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