Find Eigenvalues of 3x3 Matrix

Eigenvalues are an important concept in linear algebra and are used in a wide range of applications, including physics, engineering, and computer science. They are a measure of how a matrix stretches or shrinks a vector, and can be used to understand the behavior of linear systems. In this post, we will focus on finding the eigenvalues of a 3×3 matrix.

A 3×3 matrix is a square matrix with three rows and three columns, and can be represented as:

| a11 a12 a13 | | a21 a22 a23 | | a31 a32 a33 |

To find the eigenvalues of a 3×3 matrix, we need to solve the characteristic equation, which is given by:

det(A – λI) = 0

where A is the matrix, λ is the eigenvalue, and I is the identity matrix.

To solve this equation, we first need to subtract λI from the matrix A, where I is the identity matrix of the same size as A. For a 3×3 matrix, this gives us:

| a11-λ a12 a13 | | a21 a22-λ a23 | | a31 a32 a33-λ |

Next, we need to calculate the determinant of this matrix. The determinant of a 3×3 matrix can be calculated using the following formula:

det(A) = a11(a22a33 – a32a23) – a12(a21a33 – a31a23) + a13(a21a32 – a31a22)

Using this formula, we can calculate the determinant of the matrix (A – λI), which gives us a polynomial equation in λ. We can then solve this equation to find the eigenvalues of the matrix.

Let’s work through an example to illustrate this process. Consider the matrix:

| 2 1 1 | | 1 2 1 | | 1 1 2 |

We first need to subtract λI from this matrix, which gives us:

| 2-λ 1 1 | | 1 2-λ 1 | | 1 1 2-λ |

Next, we need to calculate the determinant of this matrix:

det(A – λI) = (2-λ)(2-λ)(2-λ) + 1(1)(2-λ) + 1(1)(1) – 1(2-λ)(1) – (2-λ)(1)(1) – 1(1)(2-λ)

Simplifying this expression, we get:

det(A – λI) = λ^3 – 6λ^2 + 9λ – 4

To find the eigenvalues, we need to solve the equation det(A – λI) = 0. In this case, we have:

λ^3 – 6λ^2 + 9λ – 4 = 0

This is a cubic equation, which can be solved using various methods, such as the cubic formula or numerical methods. In this case, we can factor the equation to get:

(λ-1)(λ-2)^2 = 0

Therefore, the eigenvalues of the matrix are λ1 = 1 and λ2 = 2 (with multiplicity 2).

In conclusion, finding the eigenvalues of a 3×3 matrix involves solving the characteristic equation det(A – λI) = 0, which gives us a polynomial equation in λ. We can then solve this equation