Cayley-Hamilton theory

Because A is a 2 x 2 matrix, from Cayley-Hamilton theory [9], A can be expanded as... 

The Linear Algebraic Problem.—Familiarity with the basic theory of finite vectors and matrices—the notions of rank and linear dependence, the Cayley-Hamilton theorem, the Jordan normal form, orthogonality, and related principles—will be presupposed. In this section and the next, matrices will generally be represented by capital letters, column vectors by lower case English letters, scalars, except for indices and dimensions, by lower case Greek letters. The vectors a,b,x,y,..., will have elements au f it gt, r) . .. the matrices A, B,...,... 

The Cayley-Hamilton theorem is one of the most powerful theorems of matrix theory. It states A matrix satisfies its own characteristic equation. That is, if the characteristic equation of an m X m matrix [A] is... 

In his Memoir on the Theory of Matrices, Cayley mentioned the important theorem for matrices, known as the Cayley-Hamilton theorem, which states that a square matrix satisfies its own characteristic polynomial. The significance of the Cayley-Hamilton theorem is that for a matrix of size n x n all information is in the first A" matrices, n = 1,... n. Thus, there is no new information to be obtained by calculating higher powers of matrices. 

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