How to find the $n^{th}$ power of $M\in\mathbb{R}^{m\times m}$, for $M$ symmetric, $m\le 1000$, and $n$ arbitrary?

Asked Oct 17 '18 at 22:12

Active Oct 10 '20 at 23:12

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I have to find the $n^{th}$ power of matrix $M$ which is symmetric about the main diagonal and all elements on the main diagonal are zero. For example: $$M=\left( \begin{array}{cc} 0 & 1 & 0 & 0\\ 1 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 \end{array} \right)$$

Here's what I've done so far:

I implemented matrix multiplication which have time complexity $O(m^3)$ and used fast exponentiation to find power of matrix in $O(\log(n))$ iterations of matrix multiplication. This has overall time complexity $O(m^3\log(n))$, which is very slow even for $m=1000$.
I find out about $n^{th}$ power of matrix on this website, which is related to eigenvalues and eigenvectors and Jordan normal form.

Now I don't know if $M$ as stated can be converted into Jordan normal form, and if so then how to find eigenvalues, eigenvectors, and finally how to use these to find $M^n$?

edited Oct 10 '20 at 23:12

Integrand

8,457

asked Oct 17 '18 at 22:12

Abhi

@Moo this matrix is sample matrix and i want to find for any matrix with given properties. – Abhi Oct 17 '18 at 22:19
how to calculate $j^m$ efficiently – Abhi Oct 17 '18 at 22:21
please it would be help how it is easy because i do not know very much about special properties of matrices – Abhi Oct 17 '18 at 22:22
... but a real symmetric matrix is diagonalizable, so no Jordan blocks needed – Robert Israel Oct 17 '18 at 22:46
Any real symmetric matrix, that is. – Robert Israel Oct 17 '18 at 22:51
@Moo I wouldn’t insist on a diagonalization via the eigensystem, though. Odds are, the eigenvalues aren’t anything “nice.” There are other ways to diagonalize a real symmetric matrix that are nicer for computation. – amd Oct 17 '18 at 22:54
i need proper steps to find nth power of matrix m which i can convert into algorithm for above properties like matrices – Abhi Oct 17 '18 at 23:58

How to find the $n^{th}$ power of $M\in\mathbb{R}^{m\times m}$, for $M$ symmetric, $m\le 1000$, and $n$ arbitrary?

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