Finding the symbolic expression for a particular matrix raised to the $n^{\text{th}}$ power Consider the following matrix, with $a$ being a real number between $0$ and $1$.
\begin{equation}
\text{M}=\left[
\begin{array}{cc}
 1-\frac{a^2}{3} & \frac{1}{3} \left(2 a-a^2\right) \\
 \frac{2 a^2}{3} & 1-\frac{2}{3}  \left(2 a-a^2\right) \\
\end{array}
\right].
\end{equation}
For a general integer $n$, I am trying to compute the closed form expression for the entries of:
\begin{equation}
\text{FinalVector}= \text{M}^{n} \left(
\begin{array}{c}
 1\\
 1 \\
\end{array}
\right).
\end{equation}

It is very easy to compute what $\text{FinalVector}$ is like for a few specific values of $n$, but is there a way to find a formula for a general $n$?
Here are the values for $\text{FinalVector}$ for the first few $n$ (from Mathematica.)
$n = 1$
$$\left(
\begin{array}{c}
 -\frac{2 a^2}{3}+\frac{2 a}{3}+1 \\
 \frac{4 a^2}{3}-\frac{4 a}{3}+1 \\
\end{array}
\right)$$
$n = 2$
$$\left(
\begin{array}{c}
 -\frac{2 a^4}{9}+\frac{10 a^3}{9}-\frac{20 a^2}{9}+\frac{4 a}{3}+1 \\
 \frac{4 a^4}{9}-\frac{20 a^3}{9}+\frac{40 a^2}{9}-\frac{8 a}{3}+1 \\
\end{array}
\right)$$
$n = 3$
$$\left(
\begin{array}{c}
 -\frac{2 a^6}{27}+\frac{2 a^5}{3}-\frac{22 a^4}{9}+\frac{122 a^3}{27}-\frac{14 a^2}{3}+2 a+1 \\
 \frac{4 a^6}{27}-\frac{4 a^5}{3}+\frac{44 a^4}{9}-\frac{244 a^3}{27}+\frac{28 a^2}{3}-4 a+1 \\
\end{array}
\right)$$
$n = 4$
$$\left(
\begin{array}{c}
 -\frac{2 a^8}{81}+\frac{26 a^7}{81}-\frac{16 a^6}{9}+\frac{440 a^5}{81}-\frac{812 a^4}{81}+\frac{308 a^3}{27}-8 a^2+\frac{8 a}{3}+1 \\
 \frac{4 a^8}{81}-\frac{52 a^7}{81}+\frac{32 a^6}{9}-\frac{880 a^5}{81}+\frac{1624 a^4}{81}-\frac{616 a^3}{27}+16 a^2-\frac{16 a}{3}+1 \\
\end{array}
\right)$$


$
\def\a{\alpha}\def\b{\beta}\def\l{\lambda}
\def\o{{\tt1}}\def\p{\partial}
\def\L{\left}\def\R{\right}\def\LR#1{\L(#1\R)}
\def\qiq{\quad\implies\quad}
\def\grad#1#2{\frac{\p #1}{\p #2}}
\def\m#1{\left[\begin{array}{c}#1\end{array}\right]}
\def\fracLR#1#2{\LR{\frac{#1}{#2}}}
$Given a rank-one matrix $A=uv^T$, multiply by $u$ and note that
$$\eqalign{
Au &= u\LR{v^Tu} = \l u
 \qiq  \l=u^Tv\quad\big({\rm eigenvalue}\big) 
}$$
Similarly, mulitply by any vector $q$ orthogonal to $v$
to find that it is also an eigenvector corresponding to an eigenvalue of $0$
$$\eqalign{
Aq &= u\LR{v^Tq} \;=\; 0
}$$
Multiplying the matrix by itself reveals an unusual property
$$\eqalign{
A^2 &= \l A,\quad A^3 &= \l^2 A,\quad\ldots\quad A^k &= \l^{k-1}A \\
}$$
which allows any analytic function of the matrix to be reduced to a
linear polynomial, i.e.
$$ f(A) = \a A + \b I $$
Combining this with the fact that the eigenvalues of $f(A)$ are given by $f(\l_k)$
generates a system of scalar equations for our two eigenvalues which can be trivially solved
for the unknown coefficients
$$\eqalign{
\a 0 + \b &= f(0) &\qiq \b = f(0) \\
\a\l + \b &= f(\l) &\qiq \a = \frac{f(\l)-f(0)}{\l} \\
}$$
Now we can write an expression for the function of a rank-one matrix,
including the limiting case when $\l\to 0\,$ (via l'Hopital's Rule)
$$\eqalign{
f(A) &=     f(0)\,I &+\; \fracLR{f(\l)-f(0)}{\l}A \\
\lim_{\l\to 0}\;f(A) &=   f(0)\,I &+\; f'(0)\;A \\
}$$
and multiply it by the all-ones vector
$$\eqalign{
f(A)\,\o &=     f(0)\,\o &+\; \fracLR{f(\l)-f(0)}{\l}\LR{v^T\o}u \\
\lim_{\l\to 0}\;f(A)\,\o &=   f(0)\,\o &+\; f'(0)\LR{v^T\o}u \\
\\
}$$

Substituting the parameters for this particular problem
$$\eqalign{
u &= \m{+\o \\ -2},
\quad &v = \frac 13\m{-a^2 \\ 2a-a^2 },
\quad A = uv^T \\
\l &= \frac{a^2-4a}3,\quad &\LR{\o+\l} = \frac{\LR{a-1}\LR{a-3}}3 \\
f(\l) &= \LR{\o+\l}^n,\quad &f'(\l) = n\LR{\o+\l}^{n-1} \\
f(0) &= \o, &f'(0) = n \\
\\
}$$
$$\eqalign{
M^n &= f(A) \\
 &= f(0)\,I \;+\; \fracLR{f(\l)-f(0)}{\l}A \\
 &= I \;+\; \fracLR{\LR{\o+\l}^n-\o}{\l}A
\qquad
\qquad
\qquad
\qquad
\quad
\\
\\
\lim_{\l\to 0}\,M^n
 &= f(0)\,I \;+\; f'(0)\;A \\
 &= I \;+\; nA \\
\\
}$$
Another interesting limit $($assuming $a\ne 0)\,$
is $\,n\to\infty$
$$\eqalign{
&0\lt a\le\o \qiq -\o\le\l\lt 0 \qiq 0\le\LR{\o+\l}\lt\o \\\\
&\lim_{n\to\infty} f(\l) = \lim_{n\to\infty}\LR{\o+\l}^n = 0 \\
&\lim_{n\to\infty} M^n = I - \fracLR{\o}{\l}A \\
}$$
