Eigenvalues and eigenvectors are important concepts in linear algebra and are used in a variety of applications, including physics, engineering, and computer science. In this article, we will discuss how to find the eigenvalues and eigenvectors of a matrix.

Definitions

Let A be an n x n matrix. An eigenvector of A is a nonzero vector v such that when A is multiplied by v, the resulting vector is a scalar multiple of v:

Av = λv

where λ is a scalar known as the eigenvalue corresponding to v. In other words, multiplying the matrix A by an eigenvector v results in a vector that points in the same direction as v, but possibly with a different magnitude.

Finding Eigenvalues and Eigenvectors

To find the eigenvalues and eigenvectors of a matrix, we start by solving the characteristic equation:

det(A – λI) = 0

where I is the identity matrix and det is the determinant function. The characteristic equation is a polynomial of degree n in λ, where n is the size of the matrix A. The roots of the characteristic equation are the eigenvalues of A.

Once we have found the eigenvalues, we can find the corresponding eigenvectors by solving the equation (A – λI)v = 0, where v is a nonzero vector. This equation is equivalent to the homogeneous system of linear equations (A – λI)x = 0, where x is a column vector obtained by stacking the components of v.

Let’s work through an example to see how this process works in practice.

Example

Consider the matrix A:

\begin{bmatrix}2 & -1 \ 4 & -3 \end{bmatrix}

To find the eigenvalues, we solve the characteristic equation:

det(A – λI) = \begin{vmatrix}2 – λ & -1 \ 4 & -3 – λ\end{vmatrix} = (2 – λ)(-3 – λ) – (-1)(4) = λ^2 – λ – 2 = 0

This is a quadratic equation that can be factored as (λ – 2)(λ + 1) = 0. Therefore, the eigenvalues of A are λ1 = 2 and λ2 = -1.

To find the eigenvectors corresponding to λ1 = 2, we solve the system of linear equations (A – 2I)v = 0:

\begin{bmatrix}0 & -1 \ 4 & -5\end{bmatrix} \begin{bmatrix}x \ y\end{bmatrix} = \begin{bmatrix}0 \ 0\end{bmatrix}

This system can be row reduced to obtain:

\begin{bmatrix}0 & -1 \ 0 & 0\end{bmatrix} \begin{bmatrix}x \ y\end{bmatrix} = \begin{bmatrix}0 \ 0\end{bmatrix}

From this, we can see that x = y, so any vector of the form v1 = \begin{bmatrix}1 \ 1\end{bmatrix} is an eigenvector corresponding to λ1 = 2.

Similarly, to find the eigenvectors corresponding to λ2 = -1, we solve the system of linear equations (A + I)v = 0:

\begin{bmatrix}3 & -1 \ 4 & -2\end{bmatrix} \begin{bmatrix}x \ y\end{bmatrix} = \begin{bmatrix}0 \ 0\end{bmatrix}

This system can be row reduced to obtain:

\begin{bmatrix}1 & -\frac{1}{3} \ 0 & 0\end{bmatrix} \begin{bmatrix}x \ y\end{bmatrix} = \begin{bmatrix}0 \ 0\end{bmatrix}

From this, we can see that x = \frac{1}{3}y, so any vector of the form v2 = \begin{bmatrix}1 \ 3\end{bmatrix} is an eigenvector corresponding to λ2 = -1.

Therefore, the eigenvalues and eigenvectors of A are:

λ1 = 2, v1 = \begin{bmatrix}1 \ 1\end{bmatrix}

λ2 = -1, v2 = \begin{bmatrix}1 \ 3\end{bmatrix}

Geometric Interpretation

The eigenvalues and eigenvectors of a matrix have a geometric interpretation. An eigenvector of a matrix represents a direction in which the matrix only stretches or compresses, without rotating. The corresponding eigenvalue represents the factor by which the vector is stretched or compressed in that direction.

In our example above, the matrix A stretches any vector in the direction of v1 by a factor of 2, and stretches any vector in the direction of v2 by a factor of -1. This can be seen by considering the action of A on the standard basis vectors e1 = \begin{bmatrix}1 \ 0\end{bmatrix} and e2 = \begin{bmatrix}0 \ 1\end{bmatrix}:

Ae1 = \begin{bmatrix}2 \ 4\end{bmatrix} = 2\begin{bmatrix}1 \ 1\end{bmatrix} = 2v1

Ae2 = \begin{bmatrix}-1 \ -3\end{bmatrix} = -1\begin{bmatrix}1 \ 3\end{bmatrix} = -1v2

In general, the eigenvectors of a matrix form a basis for the vector space, called the eigenspace of the matrix corresponding to each eigenvalue. The dimension of each eigenspace is equal to the multiplicity of the corresponding eigenvalue.

Conclusion

Eigenvalues and eigenvectors are important concepts in linear algebra that have a wide range of applications. To find the eigenvalues and eigenvectors of a matrix, we solve the characteristic equation and the corresponding system of linear equations. The geometric interpretation of eigenvectors and eigenvalues allows us to understand how a matrix stretches or compresses vectors in different directions.