# Eigenvalues of a block matrix where it has common elements

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.

• not totally clear with your matrix, does $B=D$ and $D_1\ne D$? some links that may give an idea math.stackexchange.com/questions/207865/… math.stackexchange.com/questions/21454/… – Michael Medvinsky Dec 1 '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 2 '15 at 0:01
• mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums – Michael Medvinsky Dec 2 '15 at 0:05
• 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 2 '15 at 0:10
• 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 2 '15 at 0:12