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Suppose we have a block symmetric positive definite matrix as below:

$M=\begin{bmatrix} A & B & C\\ B & E & F\\ C&F&A+D1& B&C\\&&B&E&F\\&&C&F&A+D1\\&&&&B&E&.....\\&&&&&&.....\end{bmatrix}$

Where, D1 is a diagonal matrix. So, A,E, A+D1 are all SPD. we think the above matrix as below,

$M_{First}=\begin{bmatrix} A&B&C\\B&E&F\\C&F&A/2+D1/2\end{bmatrix}$ $M_{btween}=\begin{bmatrix} A/2+D1/2&B&C\\B&E&F\\C&F&A/2+D1/2\end{bmatrix}$ $M_{Last}=\begin{bmatrix} A/2+D1/2&B&C\\B&E&F\\C&F&A\end{bmatrix}$

Also, if P1, P2 are defined as,

$P_1=\begin{bmatrix}I&&\\&I&\\&&I\\&&&\\&&&&\\&&&&...\end{bmatrix}$ $P_2=\begin{bmatrix}&&&\\&&&\\&&&I\\&&&&I\\&&&&&I\\&&&&&&\\&&&&&&...\end{bmatrix}$ similarly P3, P4, ..,

Please note that P1 and P2 has common element of identity matrix. same as P2, P3 and P4 has common element of their identity matrix.

finally it will be,

$M=P_1M_{First}P_1^T+P_2M_{between}P_2^T+...+P_iM_{between}P_i^T+..+P_NM_{Last}P_N^T$

Please note that, $M_{first}, M_{between}, M_{last}$ are all have common elements in between.

Now, the question is, how the eigenvalue and eigenvector $M_{first}$, $M_{between}$ and $M_{Last}$ are related with the whole matrix, $M$? More details as: if one can do the eigen solution of $M_{First}$, $M_{between}$ and $M_{Last}$, can one find the eigen solution of the whole.

N.B. its sparse matrix, so the blacks in the $M$, $M_{first}$, $M_{between}$ and $M_{Last}$ are zeros.

Thanks in advance, sincerely yours.

Couchy
  • 2,722
gman
  • 27
  • not totally clear with your matrix, does $B=D$ and $D_1\ne D$?

    some links that may give an idea http://math.stackexchange.com/questions/207865/the-eigenvalues-of-a-block-matrix http://math.stackexchange.com/questions/21454/prove-that-the-eigenvalues-of-a-block-matrix-are-the-combined-eigenvalues-of-its

     – Michael Medvinsky Dec 01 '15 at 23:57
  • Yes sir, B=D and here D1 is a diagonal matrix (just edited). I saw these links before posting the question, actually those solved problems have block matrices which doesn't have common elements. In the above problem, this block matrices have common elements. – gman Dec 02 '15 at 00:01
  • http://mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums – Michael Medvinsky Dec 02 '15 at 00:05
  • thanks sir, it looks interesting article. So, does it mean there is no way to find exact eigenvectors and eigenvalues of this problem? – gman Dec 02 '15 at 00:10
  • probably not in the way you asking, but other then that why not? do you need it numerically or just a theoretical question? – Michael Medvinsky Dec 02 '15 at 00:12
  • Sir, I am doing Finite Difference analysis, so numerical analysis is okay.If we can find the eigen solution of M from M_{first}, M_{between} and M_{Last}, it would save a huge amount of computational complexity (time and memory) when the matrix size is huge (over billions). That is a huge advantage. – gman Dec 02 '15 at 00:17
  • do you really need all eigenvalues? did you heard about power method? – Michael Medvinsky Dec 02 '15 at 00:25
  • Yes sir, I know the power method but I need to find all the eigen values. – gman Dec 02 '15 at 00:28
  • http://math.stackexchange.com/questions/768882/power-method-for-finding-all-eigenvectors – Michael Medvinsky Dec 02 '15 at 00:31
  • Yes, sir it is a good method, but will not the complexity go higher if the unknowns are billions? the iteration number for certain epsilon, its variable. – gman Dec 02 '15 at 01:16
  • google use power method for billions-sized matrices, but they look for the largest eigenvalue only... did you hear about dynamic computing? briefly the trick is to save results and never compute the same thing twice. if you take a simple $|M-\lambda I|$ in a block matrix form you will save a lot of work. However, the eigenvalues a usually computed using SVD. Worth to look into algorithm to see if you have a benefit of block matrix form and dynamic programming. – Michael Medvinsky Dec 02 '15 at 01:32

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