Testwiki:Reference desk/Archives/Mathematics/2016 December 20

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December 20

Power of a sparse matrix

Let $A$ be an $n\times n$ matrix, with entries $A_{0,0}=2,A_{0,1}=-1,A_{0,n-1}=1,A_{i>0,i-1}=1$, and zero elsewhere. I wish to compute the $n-1$st (last) row of $A^k$, and I don't care about the other rows. Is there a fast way (faster than naive matrix multiplication or exponentiation by squaring) to compute this? 24.255.17.182 (talk) 23:05, 20 December 2016 (UTC)

Often it is convenient to diagonalize (or put in Jordan form), then raise to powers, then un-diagonalize. I have not considered whether this is nice for your example. --JBL (talk) 23:53, 20 December 2016 (UTC)

I don't think so- I can show that the characteristic polynomial is $f_m(\lambda )=(-1{)}^m(-{\lambda }^{m+1}+2{\lambda }^m-{\lambda }^{m-1}+1)$ which doesn't factorize to anything nice in general. 24.255.17.182 (talk) 00:52, 21 December 2016 (UTC)

• You might try working out by hand a few simple examples (small n, and several small values of k) and form a hypothesis about the desired row. If the hypothesis is simple enough, it just might be easy to prove by induction. Loraof (talk) 00:19, 21 December 2016 (UTC)

You may already know this, but depending on $n$ and $k$, iterated multiplication could be faster than exponentiation by squaring. Right-multiplying a row vector by A requires only a left shift, two additions, and a cyclic permutation which can be handled as a renumbering of entries. Repeating this $k$ times takes $O(k^2)$ time and $O(nk)$ space (assuming $k>n$, ignoring factors of $\log n$, and considering that the integers grow exponentially with $k$). Exponentiation by squaring with FFT integer multiplication and $O(n^c)$ matrix multiplication would take $O(n^{c}k\log k)$ time and $O(n^{2}k)$ space, I think. Iterated multiplication also has a smaller constant factor not only because of the simple algorithm but also because of better locality of reference. -- BenRG (talk) 20:18, 23 December 2016 (UTC)