are matrix addition, multiplication by a scalar, and matrix multiplication. These are exactly those operations necessary for defining a polynomial function of an n × n matrix . If we recall from basic calculus that many functions can be written as a Maclaurin series, then we can define more general functions of matrices quite easily. If is diagonalizable, that is with then and the evaluation of the Maclaurin series for functions of is greatly simplified. For example, to obtain any power k of , we need only compute , premultiply by , and postmultiply the result by . Using generalized eigenvectors, we can obtain the Jordan normal form for and these results can be generalized to a straightforward method for computing functions of nondiagonalizable matrices. (See Matrix function#Jordan decomposition.) Differential equations Consider the problem of solving the system of linear ordinary differential equations where and If the matrix is a diagonal matrix so that for , then the system () reduces to a system of n equations which take the form In this case, the general solution is given by In the general case, we try to diagonalize and reduce the system () to a system like () as follows. If is diagonalizable, we have , where is a modal matrix for . Substituting , equation () takes the form , or where The solution of () is The solution of () is then obtained using the relation (). On the other hand, if is not diagonalizable, we choose to be a generalized modal matrix for , such that is the Jordan normal form of . The system has the form where the are the eigenvalues from the main diagonal of and the are the ones and zeros from the superdiagonal of . The system () is often more easily solved than (). We may solve the last equation in () for , obtaining . We then substitute this solution for into the next to last equation in () and solve for . Continuing this procedure, we work through () from the last equation to the first, solving the entire system for . The solution is then obtained using the relation (). Lemma: Given the following chain of generalized eigenvector of length r as: These functions solve the system of equations: Proof: let define the following sum: Then: on the other hand we have: as required. Notes References Linear algebra