1.8.1 Eigenvalues & Eigenvectors

Characteristic Polynomials

Eigenvalues and eigenvectors are properties of square matrices and are used in a lot of real-life applications including geometrical transformations and probability scenarios. In order to find these eigenvalues and eigenvectors, the characteristic polynomial for a matrix must be found and solved.

What is a characteristic polynomial?

For a matrix $A$ , if $A x = λ x$ when $x$ is a non-zero vector and $λ$ a constant, then $λ$ is an eigenvalue of the matrix $A$ and $x$ is its corresponding eigenvector
If $A x = λ x \Rightarrow (λ I - A) x = 0$ or $(A - λ I) x = 0$ and for $x$ to be a non-zero vector, $\det (λ I - A) = 0$
The characteristic polynomial of an $n \times n$ matrix is:

$p (λ) = \det (λ I - A)$

In this course you will only be expected to find the characteristic equation for a $2 \times 2$ matrix and this will always be a quadratic

How do I find the characteristic polynomial?

STEP 1
Write $λ I - A$ , remembering that the identity matrix must be of the same order as $A$
STEP 2
Find the determinant of $λ I - A$ using the formula given to you in the formula booklet

$\det A = | A | = a d - b c$

STEP 3
Re-write as a polynomial

Exam Tip

You need to remember the characteristic equation as it is not given in the formula booklet

Worked example

Find the characteristic polynomial of the following matrix

$A = (\begin{matrix} 5 & 4 \\ 3 & 1 \end{matrix})$ .

1-8-1-ib-ai-hl-eigenvalues--eigenvectors-we-1-solution

Eigenvalues & Eigenvectors

How do you find the eigenvalues of a matrix?

The eigenvalues of matrix $A$ are found by solving the characteristic polynomial of the matrix
For this course, as the characteristic polynomial will always be a quadratic, the polynomial will always generate one of the following:

two real and distinct eigenvalues,
one real repeated eigenvalue or
complex eigenvalues

How do you find the eigenvectors of a matrix?

A value for $x$ that satisfies the equation is an eigenvector of matrix $A$
Any scalar multiple of $x$ will also satisfy the equation and therefore there an infinite number of eigenvectors that correspond to a particular eigenvalue
STEP 1
Write $x = (\begin{matrix} x \\ y \end{matrix})$
STEP 2
Substitute the eigenvalues into the equation $(λ I - A) x = 0$ , and form two equations in terms of $x$ and $y$
STEP 3
There will be an infinite number of solutions to the equations, so choose one by letting one of the variables be equal to $1$ and using that to find the other variable

Exam Tip

You can do a quick check on your calculated eigenvalues as the values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix