Linear Algebra: Find a formula for a matrix M to any power

[Neutrinogun]

Homework Statement

Homework Equations

A = SDS^-1

Under some specific conditions,

Aⁿ=SDⁿS^-1

The Attempt at a Solution

det(A-λI) = 0
(16-λ)(-λ) - (-64)(1) = 0
λ² - 16λ + 64 = 0

λ = 8 Multiplicity 2.

This is as far as I got because you need 2 eigenvalues to get 2 eigenvectors to create the S/D matrices. Is there another way to solve this, because this is the only way I learned. Also there has to be an answer for this because it is on webwork with fill in the blank answers.

 

[Ray Vickson]

Homework Statement

Homework Equations

A = SDS^-1

Under some specific conditions,

Aⁿ=SDⁿS^-1

The Attempt at a Solution

det(A-λI) = 0
(16-λ)(-λ) - (-64)(1) = 0
λ² - 16λ + 64 = 0

λ = 8 Multiplicity 2.

This is as far as I got because you need 2 eigenvalues to get 2 eigenvectors to create the S/D matrices. Is there another way to solve this, because this is the only way I learned. Also there has to be an answer for this because it is on webwork with fill in the blank answers.

For an $n \times n$ matrix $A$, if $r_1, r_2, \ldots, r_p$ are the distinct eigenvalues of multiplicities $m_1, m_2, \ldots, m_p$, it follows from the Jordan canonical form that there are matrices $E_{i,k_i}, i = 1, \ldots, p, k_i = 1, \ldots, m_i$ such that
$$P(A) = \sum_{i=1}^p [E_{i1} P(r_i) + E_{i2} P'(r_i) + \cdots + E_{i,m_i} P^{(m_i-1)}(r_i) ]$$
The matrices $E_{ik}$ are fixed, and are the same for all functions $P$.

In your case, $P(M) = E_1 P(8) + E_2 P'(8)$, where the matrices $E_1, E_2$ are the same for any polynomial $P$ (and, in fact, for any analytic function $f(M)$). You can get them from two known values $P(M)$, such as for $P(x) = 1 \Longrightarrow P(M) = I$ (the identity matrix) and $P(x) = x \Longrightarrow P(M) = M$. For $P(x) = 1, P'(x) = 0$ we get $I = E_1 + 0 E_2$ and for $P(x) = x, P'(x) = 1$ we get $M = 8 E_1 + 1 E_2$. You can solve for $E_1, E_2$, then get $M^n = E_1 8^n + E_2\, n 8^{n-1}$.