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 \( A \).


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