Eigenvector

The eigenvector is a vector that is associated with a set of linear equations. The eigenvector of a matrix is also known as a latent vector, proper vector, or characteristic vector. These are defined in the reference of a square matrix. Eigenvectors are also useful in solving differential equations and many other applications related to them. In this article, let us discuss the eigenvector definition, equation, methods with examples in detail.

Eigenvector Definition

Eigenvector of a square matrix is defined as a non-vector in which when a given matrix is multiplied,  it is equal to a scalar multiple of that vector. Let us suppose that A is an n x n square matrix, and if v be a non-zero vector, then the product of matrix A, and vector v is defined as the product of a scalar quantity λ and the given vector, such that:

Av =λv

Where

v = Eigenvector and λ be the scalar quantity that is termed as eigenvalue associated with given matrix A

Eigenvector Equation

The equation corresponding to each eigenvalue of a matrix is given by:

AX = λ X

It is formally known as the eigenvector equation.

In place of λ, substitute each eigenvalue and get the eigenvector equation which enables us to solve for the eigenvector belonging to each eigenvalue.

Eigenvector Method

The method of determining the eigenvector of a matrix is given as follows:

If A be an n×n matrix and λ be the eigenvalues associated with it. Then, eigenvector v can be defined by the following relation: Av =λv

If “I” be the identity matrix of the same order as A, then

(A – λI)v =0

The eigenvector associated with matrix A can be determined using the above method.

Here, “v” is known as eigenvector belonging to each eigenvalue and is written as:

\(v =\begin{bmatrix} v_{1}\\ v_{2}\\ .\\ .\\ v_{n}\end{bmatrix}\)

How to Find an Eigenvector?

To find the eigenvectors of a matrix, follow the procedure given below:

  1. Find the eigenvalues of the given matrix A, using the equation det ((A – λI) =0, where “I” is equivalent order identity matrix as A. Denote each eigenvalue of λ1, λ2, λ3….
  2. Substitute the values in the equation AX = λ1 or (A – λ1 I) X = 0.
  3. Calculate the value of eigenvector X, which is associated with the eigenvalue.
  4. Repeat the steps to find the eigenvector for the remaining eigenvalues.

Types of Eigenvector

The eigenvectors are of two types namely,

  • Left Eigenvector
  • Right Eigenvector

Left Eigenvector

The left eigenvector is represented in the form of a row vector which satisfies the following condition:

AXL=λXL

Where

A is a given matrix of order n and λ be one of its eigenvalues.

XL is a row vector of a matrix. I,e., [ x1 x2 x3 …. Xn]

Right Eigenvector

The right eigenvector is represented in the form of a column vector which satisfies the following condition:

AXR=λXR

Where

A is a given matrix of order n and λ be one of its eigenvalues.

XR is a column vector of a matrix. I,e., \(X_{R} =\begin{bmatrix} x_{1}\\ x_{2}\\ .\\ .\\ x_{n}\end{bmatrix}\)

Eigenvector Applications

The important application of eigenvectors are as follows:

  • Eigenvectors are used in Physics in simple mode of oscillation
  • In Mathematics, eigenvector decomposition is widely used in order to solve the linear equation of first order, in ranking matrices, in differential calculus etc
  • This concept is widely used in quantum mechanics
  • It is applicable in almost all the branches of engineering

Eigenvector Examples

Example: Find the eigenvector of the given matrix:

\(A =\begin{bmatrix} 1 &4 \\ -4 & -7 \end{bmatrix}\)

Solution:

Given: \(A =\begin{bmatrix} 1 &4 \\ -4 & -7 \end{bmatrix}\)

 

\(|A – \lambda I| =\begin{vmatrix} 1-\lambda & 4\\ -4& -7-\lambda \end{vmatrix}\)

(1- λ)(-7- λ)- 4(-4) = 0

( λ+3)2 = 0

Therefore, λ =-3, -3

Use the eigenvector equation

AX = λX

Substitute λ value in the equation:

AX = -3X

We know that,

(A- λI) X = 0

\(\left ( \begin{bmatrix} 1 &4 \\ -4&-7 \end{bmatrix} + \begin{bmatrix} 3 &0 \\ 0 & 3 \end{bmatrix} \right )\begin{bmatrix} x\\ y\end{bmatrix} = \begin{bmatrix} 0\\ 0\end{bmatrix}\)

4x +4y =0

Or

x+y =0

Assume that x =k

So, it becomes

k +y =0

y= -k

Therefore, the eigenvector is

\(X=\begin{bmatrix} x\\ y\end{bmatrix}=k\begin{bmatrix} 1\\ -1\end{bmatrix}\)

Now, let’s understand how we can find the eigenvalue of the matrix along with a solved example here.

Eigenvalue of Matrix

Eigenvalues are generally associated with eigenvectors in Linear algebra. Both of these terms are used in the interpretation of linear transformations. As we know that, eigenvalues are the particular set of scalar values related to linear equations, most probably in the matrix equations. 

To define eigenvalues, first, we have to determine eigenvectors. Almost all vectors change their direction when they are multiplied by A. Some rare vectors say x is in the same direction as Ax. These are the “eigenvectors”. Multiply an eigenvector by A, and the vector Ax is the number time of the original x. The basic equation is given by:

Ax = λx. 

Here, the number λ is an eigenvalue of matrix A.

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Frequently Asked Questions – FAQs

What are eigenvectors used for?

Eigenvalues and eigenvectors are used to reduce a linear operation to separate or simplify the problems.

How do you find eigenvectors?

The below steps help in finding the eigenvectors of a matrix.
Step 1: Find the eigenvalues of the given matrix A, using the equation det ((A – λI) =0, where “I” is an identity matrix of equivalent order as A.
Step 2: Denote each eigenvalue of λ_1, λ_2, λ_3,…
Step 3: Substitute the values in the equation AX = λ1 or (A – λ1 I) X = 0.
Step 4: Calculate the value of eigenvector X, which is associated with the eigenvalue.
Step 5: Repeat the steps to find the eigenvector for the remaining eigenvalues.

What do eigenvalues and eigenvectors tell us?

An eigenvalue is a number that tells us how much variance exists in the data in that direction, whereas an eigenvalue is a number that tells us how spread of the data.

Are eigenvectors orthogonal?

Generally, for any matrix, the eigenvectors are not always orthogonal. However, they will be orthogonal for a particular type of matrix such as a symmetric matrix.

Are all eigenvectors linearly independent?

Eigenvectors are linearly independent when the corresponding eigenvalues of a matrix are distinct.

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