# Matrix decomposition

I have a symmetric matrix $V$ of order 10. I want to decompose $V$ in such a way that $$V=SS'$$ with $S$ being non triangular. The matrix $S$ has some restrictions that 45 cells have ‘0’ values and 55 other remaining cells are supposed to be solved.

I am looking for solving this using sas.

Here is the structure of $S$ matrix (all 1's are to be replaced by variables; 55 variables and 45 zeros) $$\begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \\ \end{bmatrix}$$

• Thanks Glen. I would require this while implementing SVAR (structural VAR) restrictions. – rohitbernanke Apr 9 '14 at 9:49
• Is the pattern of zeros such that permutations of the rows and/or columns could yield a triangular matrix? (I think it's sufficient if there are rows and columns that have no zeros, 1 zero, ... 9 zeros) – Glen_b Apr 9 '14 at 10:38
• The pattern of restriction in S is such that, we cant make it in form of triangular matrix. Had it been triangular one we would have used Cholesky decomposition. – rohitbernanke Apr 9 '14 at 11:09
• What sort of pattern do you have? – Glen_b Apr 9 '14 at 11:18
• @Glen_b I have added the pattern of restrictions in the question. – rohitbernanke Apr 9 '14 at 11:44