# Eigenvalues – Definition With Examples

Created on Dec 26, 2023

Updated on January 5, 2024

Welcome to Brighterly, your trusted guide in the world of mathematics learning. In this article, we dive into the concept of eigenvalues, focusing on 2×2 and 3×3 matrices. At Brighterly, we believe in making complex mathematical concepts accessible and understandable for students of all ages. Here, you’ll learn how to calculate eigenvalues step-by-step, enhancing your algebra skills and deepening your mathematical understanding.

## Eigenvalues

Eigenvalues are a fundamental concept in mathematics, especially in linear algebra. They play a crucial role in understanding the behavior of linear transformations and matrices. An eigenvalue is a special number associated with a square matrix. It represents the scalar value` λ (lambda)`

for which there exists a non-zero vector `v`

, such that when the matrix multiplies vector v, the product is the same as when` λ multiplies v`

. This relationship can be represented by the equation `$Av=λv$`

, where `A`

is the matrix.

Understanding eigenvalues helps in various mathematical and real-world applications, such as solving differential equations, understanding rotations and stretches of space, and in more advanced fields like quantum mechanics and stability analysis.

## Calculating Eigenvalues

To calculate the eigenvalues of a matrix, we need to find the values of λ for which the equation `$Av=λv$`

holds true. This involves solving the characteristic equation of the matrix, which is derived from the determinant of the matrix `$A−λI$`

, where I is the identity matrix of the same size as `A`

. The characteristic equation is typically a polynomial, and the roots of this polynomial are the eigenvalues. The process of finding these values involves algebraic manipulation and solving polynomial equations.

## Eigenvalues of a 2×2 Matrix

When dealing with a 2×2 matrix, calculating its eigenvalues involves solving a quadratic equation derived from the matrix’s characteristic equation. This characteristic equation is obtained by setting the determinant of the matrix minus `λ`

times the identity matrix equal to zero. The general form of the equation is `$λ_{2}−trace(A)λ+det(A)=0$`

, where the trace of `A`

is the sum of its diagonal elements, and `det(A)`

is its determinant.

For example, consider a generic 2×2 matrix with elements `a`

, `b`

, `c`

, and `d`

. The characteristic equation would be `$λ_{2}−(a+d)λ+(ad−bc)=0$`

. Solving this quadratic equation provides the two eigenvalues of the matrix. These eigenvalues are key in understanding the matrix’s properties and behavior in various mathematical contexts.

## Eigenvalues of a 3×3 Matrix

The calculation of eigenvalues for a 3×3 matrix is somewhat more complex than for a `2x2`

matrix, as it involves solving a cubic polynomial. The characteristic equation in this case is also derived from the determinant of the matrix minus `λ`

times the identity matrix, but now the determinant calculation and the polynomial equation are more involved due to the increased size of the matrix.

To illustrate, for a `3x3`

matrix, the characteristic equation can be written in the form `$λ_{3}−a_{1}λ_{2}+a_{2}λ−a_{3}=0$`

, where $a_{1},a_{2},$ and a$_{3}$ are coefficients calculated based on the elements of the matrix. The roots of this cubic polynomial are the eigenvalues of the `3x3`

matrix. These eigenvalues are crucial for understanding the matrix’s linear transformations and their effects in higher-dimensional spaces.

The process of solving these equations, especially for the 3×3 matrix, requires a good grasp of algebraic methods and can be more challenging. However, it provides valuable insights into the matrix’s characteristics and applications in various mathematical and practical fields.

## Properties of Eigenvalues

Eigenvalues have several interesting properties. For instance, the sum of eigenvalues of a matrix equals the trace of the matrix (the sum of the diagonal elements), and the product of the eigenvalues equals the determinant of the matrix. These properties are useful for quickly assessing certain characteristics of the matrix without computing all the eigenvalues. Additionally, if a matrix is symmetric, its eigenvalues are always real numbers.

## Applications of Eigenvalues

Eigenvalues have wide-ranging applications in various fields of science and engineering. They are used in physics for understanding systems’ stability, in computer science for algorithms in graph theory, and in economics for analyzing economic models. In vibration analysis, eigenvalues help determine the natural frequencies at which structures like buildings and bridges resonate.

## Practice Exercises on Eigenvalues

To master the concept of eigenvalues, practicing with various types of matrices is essential. Exercises can range from calculating the eigenvalues of given matrices to more complex tasks involving the application of eigenvalues in different contexts. Regular practice helps in understanding the nuances of the concept and its applications.

## Frequently Asked Questions on Eigenvalues

**What is an eigenvalue?**

An eigenvalue is a special scalar associated with a square matrix, indicating a value for which there exists a non-zero vector that, when multiplied by the matrix, results in the vector being scaled by that scalar.

**How are eigenvalues calculated?**

Eigenvalues are calculated by solving the characteristic equation derived from the determinant of `$A−λI$`

, where A is the matrix and I is the identity matrix of the same size.

**Why are eigenvalues important?**

Eigenvalues are important as they provide insights into the properties of linear transformations represented by matrices and have numerous applications in science, engineering, and mathematics.