$A$ is square symmetric matrix as:

$$A=\begin{pmatrix} A_1&A_2 \\ A_3 & A_4 \end{pmatrix}$$ I have two points which need help to understand clearly:
All blocks $A_1$, $A_2$, $A_3$, $A_4$ of symmetric $A$ are symmetric too. Is it right?
Is any relation between eigenvectors/eigenvalues of matrix $A$ and its blocks?
Thank you for your help.

  • $\begingroup$ I don't understand, what $A$ is ? $\endgroup$ – Belgi Aug 7 '12 at 8:52
  • $\begingroup$ I've edited your post using TeX syntax for better readability. Please, check whether I did unintentionally not change the meaning (and edit the post again, if needed). For some basic information about writing math at this site see e.g. here, here and here. $\endgroup$ – Martin Sleziak Aug 7 '12 at 8:55
  • $\begingroup$ Ohh, I'm seeking a editing, and see your correction. Thanks to Martin Sleziak! $\endgroup$ – HongTu Aug 7 '12 at 9:00

$A_1$ and $A_4$ must be symmetric, but $A_2$ and $A_3$ need not (in fact they're not even necessarily square); the condition here is $A_2 = ^tA_3$. Consider:

$$A = \left( \begin{array}{c|cc} 4 & 1 & 2 \\ \hline 1 & 1 & 0 \\ 2 & 0 & 1 \end{array} \right)$$

I'm not aware of any relation between the eigenvalues of $A$ and the ones of its blocks, for example here $A_1$ has eigenvalue 4, but $A$ itself doesn't have this eigenvalues. As for the eigenvectors, they don't even belong in the same spaces (they're 3-dimensional for $A$ but not for the blocks).

  • $\begingroup$ If there is one more condition: A has size nxn, n is even number, the first thing is right - all blocks are symmetric; A_2 and A_3 are equal? $\endgroup$ – HongTu Aug 7 '12 at 9:27

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