# The characteristic equation

### The characteristic equation

#### Lessons

We say that a scalar $\lambda$ is an eigenvalue of an $n \times n$ matrix $A$ if and only if $\lambda$ satisfies the following characteristic equation:
$\det(A-\lambda I)=0$

We say that $\det(A-\lambda I)$ is a characteristic polynomial.

Useful ways to find eigenvalues
When dealing with a $2 \times 2$ matrix, use the formula $\det (A) = ad-bc$.

When dealing with a $3 \times 3$ matrix, use the shortcut method.

When dealing with a triangular matrix, know that the determinant is just the product of the diagonal entries.

Note that:
1. An eigenvalue is a distinct root if it has a multiplicity of 1
2. An eigenvalue is a repeated root if it has a multiplicity greater than 1
• Introduction
The Characteristic Equation Overview:
a)
What is the Characteristic Equation?
$\det (A-\lambda I)=0$
• The characteristic polynomial is $\det (A-\lambda I)$

b)
Finding the Eigenvalue
• The eigenvalues of a $2 \times 2$ matrix
• The eigenvalues of a $3 \times 3$ matrix
• The eigenvalues of a triangular matrix

c)
Shortcut to Determinants of Matrices
$2 \times 2$ matrices
$3 \times 3$ matrices
• Triangular matrices

d)
Eigenvalue with Multiplicity
• Distinct eigenvalues
• Repeated eigenvalues

• 1.
Finding the Characteristic Polynomial
Find the characteristic polynomial of $A$ if:

• 2.
Find the characteristic polynomial of $A$ if:

• 3.
Finding the Eigenvalues of a $2 \times 2$ matrix
Find all the eigenvalues of the matrix
State their multiplicities, and what type of eigenvalues they are.

• 4.
Finding the Eigenvalues of a $3 \times 3$ matrix
Find all the eigenvalues of the matrix
State their multiplicities, and what type of eigenvalues they are.

• 5.
Finding the Eigenvalues of a triangular matrix
Find all the eigenvalues of the matrix
State their multiplicities, and what type of eigenvalues they are.

• 6.
Proofs dealing with the Characteristic Equation
Show that $A$ and $A^T$ has the same characteristic polynomials.

• 7.
Suppose $A$ is an $n \times n$ triangular matrix where all the diagonal entries are $c$. Then the characteristic polynomial is
$p(\lambda)=(c-\lambda)^n$