Diagonalizable Matrix - An Application

An Application

Diagonalization can be used to compute the powers of a matrix A efficiently, provided the matrix is diagonalizable. Suppose we have found that

is a diagonal matrix. Then, as the matrix product is associative,

\begin{align} A^k &= (PDP^{-1})^k = (PDP^{-1}) \cdot (PDP^{-1}) \cdots (PDP^{-1}) \\ &= PD(P^{-1}P) D (P^{-1}P) \cdots (P^{-1}P) D P^{-1} \\ &= PD^kP^{-1} \end{align}

and the latter is easy to calculate since it only involves the powers of a diagonal matrix. This approach can be generalized to matrix exponential and other matrix functions since they can be defined as power series.

This is particularly useful in finding closed form expressions for terms of linear recursive sequences, such as the Fibonacci numbers.

Read more about this topic:  Diagonalizable Matrix

Famous quotes containing the word application:

    “If you would be a favourite of your king, address yourself to his weaknesses. An application to his reason will seldom prove very successful.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    “The main object of a revolution is the liberation of man ... not the interpretation and application of some transcendental ideology.”
    —Jean Genet (1910–1986)