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:

    “The best political economy is the care and culture of men; for, in these crises, all are ruined except such as are proper individuals, capable of thought, and of new choice and the application of their talent to new labor.”
    —Ralph Waldo Emerson (1803–1882)

    ““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)