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:

    We will not be imposed upon by this vast application of forces. We believe that most things will have to be accomplished still by the application called Industry. We are rather pleased, after all, to consider the small private, but both constant and accumulated, force which stands behind every spade in the field. This it is that makes the valleys shine, and the deserts really bloom.
    Henry David Thoreau (1817–1862)

    My business is stanching blood and feeding fainting men; my post the open field between the bullet and the hospital. I sometimes discuss the application of a compress or a wisp of hay under a broken limb, but not the bearing and merits of a political movement. I make gruel—not speeches; I write letters home for wounded soldiers, not political addresses.
    Clara Barton (1821–1912)