Monday, February 10, 2014
A Little Matrix Analysis and Ring Theory
The partitioning of the transition matrix for the cat and mouse problem into a simple matrix sum and the multinomial expression for its powers made use of matrix analysis. A lot of the mathematics of linear algebra was developed during the 19th Century. The subject has grown more abstract over time and can be difficult for the uninitiated to follow. For a better understanding of the subject one can study the history of ring theory, Clifford algebras, the Cayley-Hamilton theorem, etc. The Cayley-Hamilton theorem allows one to write a matrix power as a polynomial of its lower powers. It applies if the order of the multiplications doesn't matter. The multinomial formula for the powers of P made use of the observation that the set of matrices U, V, X and W were closed under multiplication. The Cayley-Hamilton theorem doesn't apply to P in this case because of the cross product X but it does apply to U and V separately. Including X in the ring is not a major difficulty.