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:

    “May my application so close
    To so endless a repetition
    Not make me tired and morose
    And resentful of man’s condition.”
    —Robert Frost (1874–1963)

    “Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.”
    —Talcott Parsons (1902–1979)