# Eigenvalues and Eigenvectors

## Introduction

Definition 1

Assume that $$L:\mathbb R^k\to\mathbb R^k$$ is a linear operator. If the vector $$v\in\mathbb R^k$$ and the scalar $$\lambda\in\mathbb R$$ satisfy $$L v=\lambda v$$, then $$v$$ is called an eigenvector of $$L$$. The scalar $$\lambda$$ is called an eigenvalue of $$L$$.

Clearly, zero vector is always an eigen-vector. Also, if $$u$$ is an eigenvector, then $$\kappa u$$ is also an eigenvector for every $$\kappa\in\mathbb R$$. Indeed, assuming that $$\lambda$$ is the eigenvalue corresponding to $$u$$ we have $$A(\kappa u)=\kappa A(u)=\kappa\lambda u=\lambda \kappa u$$.

Example 1 Find the eigenvalues and the eigenvectors of the operator with the matrix \begin{eqnarray*}A&=&\left[\begin{array}{cc}5&4\\-4&-5\end{array}\right]. \end{eqnarray*}

In the previous example, we found eigenvalues as the zeroes of the polynomial $$\varphi_A(\lambda)=\lambda^2-9$$. This is called the characteristic polynomial of the matrix $$A$$. More precisely,

Definition 2

Let $$A$$ be an $$n\times n$$ matrix. The polynomial $$\displaystyle \varphi_A(\lambda)=\mbox{det }\left(A-\lambda I\right)$$ is called the characteristic polynomial of the matrix $$A$$.

The proof of the following theorem is obvious once we have seen the solution of Example 1.

Theorem 1

Assume that $$A$$ is an $$n\times n$$ matrix. A real number $$\eta$$ is an eigenvalue of $$A$$ if and only if $$\varphi_A(\eta)=0$$.

## Polynomials with matrices

We will use the eigenvectors and eigenvalues to find closed formulas for $$n$$-th powers of matrices. We will illustrate the method by considering the following example.

Example 2

Let $$\displaystyle A=\left[\begin{array}{cc} 5&4\\-4&-5\end{array}\right]$$. Let us denote by $$a_n$$, $$b_n$$, $$c_n$$, and $$d_n$$ the numbers such that $$\displaystyle A^n=\left[\begin{array}{cc} a_n&b_n\\c_n&d_n\end{array}\right]$$. Find the formulas for $$a_n$$, $$b_n$$, $$c_n$$, and $$d_n$$.

Theorem 2

Assume that $$A$$ is an $$n\times n$$ matrix that has $$n$$ linearly independent eigenvectors $$v_1$$, $$\dots$$, $$v_n$$. Assume that $$\lambda_1$$, $$\dots$$, $$\lambda_n$$ are eigenvalues corresponding to $$v_1$$, $$\dots$$, $$v_n$$. Then there exists an invertible $$n\times n$$ matrix $$P$$ such that $P^{-1}AP=\left[\begin{array}{ccccc} \lambda_1&0&0&\cdots&0\\0&\lambda_2&0&\cdots&0\\ 0&0&\lambda_3&\cdots&0\\&&&\vdots&\\ 0&0&0&\dots&\lambda_n\end{array}\right].$

Theorem 3 (Cayley-Hamilton)

Assume that $$A$$ is an $$n\times n$$ matrix and $$\varphi_A$$ its characteristic polynomial. Then $$\varphi_A(A)=0$$.

## Recursive systems of equations

Our next goal is to use the techniques of eigenvalues and eigenvectors to solve the recursive systems of equations.

Example 3

Assume that $$(x_n)_{n=0}^{\infty}$$ and $$(y_n)_{n=0}^{\infty}$$ are two sequence of real numbers defined in the following way: $$x_0=3$$, $$y_0=2$$, and \begin{eqnarray*} x_{n+1}&=&5x_n+4y_n\\ y_{n+1}&=&-4x_n-5y_n, \end{eqnarray*} for $$n\geq 0$$. Determine the formulas for $$x_n$$ and $$y_n$$.

Using the technique described above we can solve the recursive equations. The following example provides the formula for Fibonacci numbers.

Example 4 (Fibonacci numbers)

Assume that $$(F_n)_{n=0}^{\infty}$$ is the sequence defined as $$F_0=0$$, $$F_1=1$$ and for $$n\geq 0$$ the following equation holds: $F_{n+2}=F_{n+1}+F_n.$ Prove that $F_n=\frac1{\sqrt 5}\left(\frac{1+\sqrt 5}2\right)^n-\frac1{\sqrt 5}\left(\frac{1-\sqrt 5}2\right)^n.$

In the next example we treat the recursive system of equations whose matrix does not have a basis of eigenvectors. This is an introductory example to Jordan forms of matrices.

Example 5

Consider the matrix $$\displaystyle A=\left[\begin{array}{cc}4&1\\-1&2\end{array}\right]$$ and the following system of equations: \begin{eqnarray*} x_{n+1}&=&4x_n+y_n\\ y_{n+1}&=&-x_n+2y_n, \end{eqnarray*} with the initial conditions $$x_0=2$$, $$y_0=5$$.

• (a) Prove that $$A$$ has only one eigenvalue $$\lambda$$ and determine $$\lambda$$.

• (b) Find an eigenvector $$u$$ corresponding to $$\lambda$$.

• (c) Does there exist an eigenvector $$w$$ of $$A$$ such that $$u$$ and $$w$$ are not scalar multiples of each other?

• (d) Find a vector $$v$$ such that $$Av=\lambda v+u$$. Here $$u$$ and $$\lambda$$ are the eigenvector and the eigenvalue from the previous parts of the problem.

• (e) Determine the matrix $$A^n$$.

• (f) Find the closed formulas for $$x_n$$ and $$y_n$$.

Remark. The vectors $$u$$ and $$v$$ from previous example form a basis called Jordan basis for the matrix $$A$$.

## Practice problems

Problem 1. Let $$A$$ be the matrix given by: \begin{eqnarray*}A=\left[\begin{array}{ccc} 6 & 5 \\ 30 & 11 \end{array}\right].\end{eqnarray*} What is the eigenvalue corresponding to the eigenvector $$\displaystyle w=\left[\begin{array}{c}1\\-2\end{array}\right]$$?

Problem 2. Assume that $$(x_n)_{n=0}^{\infty}$$ and $$(y_n)_{n=0}^{\infty}$$ are two sequences that for all $$n\geq 0$$ satisfy the following equations: \begin{eqnarray*} x_{n+1}&=&9x_n+5y_n\\ y_{n+1}&=&6y_n-8x_n. \end{eqnarray*} Let $$\displaystyle A=\left[\begin{array}{ccc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right]$$ be the matrix such that $$\displaystyle \left[\begin{array}{c} x_{n+1}\\ y_{n+1}\end{array}\right]= A\left[\begin{array}{c} x_{n}\\ y_{n}\end{array}\right]$$. Determine $$a_{11}+a_{22}$$.

Problem 3. Let $$A$$ be the matrix given by: \begin{eqnarray*}A=\left[\begin{array}{cc} 7 & -3 \newline 10 & -4 \end{array}\right]\end{eqnarray*} and let $$w_1$$, $$w_2$$, $$w_3$$, $$w_4$$, $$w_5$$ be the vectors given by: \begin{eqnarray*} w_1=\left[\begin{array}{c}3\newline5\end{array}\right],\quad w_2=\left[\begin{array}{c}5\newline7\end{array}\right],\quad w_3=\left[\begin{array}{c}1\newline2\end{array}\right],\quad w_4=\left[\begin{array}{c}1\newline3\end{array}\right],\quad w_5=\left[\begin{array}{c}1\newline5\end{array}\right]. \end{eqnarray*} Which of the above five vectors are the eigenvectors of the matrix $$A$$?

Problem 4. Let $$A$$ be the matrix given by: \begin{eqnarray*}A=\left[\begin{array}{ccc} 2 & -3 & 5\newline -1 & 10 & -14\newline -2 & 4 & -6 \end{array}\right]\end{eqnarray*} and let $$w_1$$, $$w_2$$, $$w_3$$, $$w_4$$, $$w_5$$ be the vectors given by: \begin{eqnarray*} w_1=\left[\begin{array}{c}0\newline3\newline2\end{array}\right],\quad w_2=\left[\begin{array}{c}-2\newline5\newline3\end{array}\right],\quad w_3=\left[\begin{array}{c}-7\newline19\newline10\end{array}\right],\quad w_4=\left[\begin{array}{c}-1\newline5\newline17\end{array}\right],\quad w_5=\left[\begin{array}{c}-1\newline3\newline2\end{array}\right]. \end{eqnarray*} Which three of the above five vectors are the eigenvectors of the matrix $$A$$?

Problem 5.

Which of the following statements are true?

• (S1) Every $$2\times 2$$ matrix $$A$$ has an eigenvector.

• (S2) If $$A$$ and $$B$$ are two $$n\times n$$ matrices such that $$AB=BA$$, and if $$u$$ is an eigenvector of $$B$$, then $$Au$$ is an eigenvector of $$B$$ as well.

• (S3) If $$u$$ and $$v$$ are eigenvectors of a $$2\times 2$$ matrix $$A$$, then $$u-v$$ is also an eigenvector of $$A$$.