Eigenvectors of a Matrix

Introduction

Eigenvector of a Matrix is also known as a Proper Vector, Latent Vector or Characteristic Vector. Eigenvectors are defined as a reference of a square matrix. A matrix represents a rectangular array of numbers or other elements of the same kind. It generally represents a system of linear equations. 


Eigen Vectors is a very useful concept related to matrices. These are also used in calculus to solve differential equations and many other applications related to it. There are basically two types of eigenvectors:

  1. Left Eigenvector

  2. Right Eigenvector

Let us go ahead and understand the eigenvector, how to find the eigenvalue of a 2×2 matrix, its technique and various other concepts related to it.


Eigenvector Method

The method of determining the eigenvector of a matrix is explained below:

If A be an n×n matrix and λ (lambda) be the eigenvalues associated with it. Then, eigenvector v can be defined as:

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 corresponds to eigenvector belonging to each eigenvalue and is written as:

v = \[\begin{bmatrix} v_{1}\\ v_{2}\\ . \\ . \\ . \\ v_{n} \end{bmatrix}\]


Eigenvector Equation

The equation corresponding to each eigenvalue of a matrix can be written as:

AX = λ X

It is formally known as the eigenvector equation.

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

For example: Suppose that there are two eigenvalues λ1​ = 0 and λ2​ = 1 of any 2×2 matrix.

Then,

AX = λ1​ X A =  O …..(1)

and

AX = λ2​ X A =  1

(A–I)X = O…. (2)

Equations (1) and (2) are eigenvector equations for a given matrix.

Where I corresponds to the identity matrix of the same order as A

O = zero matrix of the same order as A X = Eigenvector which is equal to \[\begin{bmatrix} x \\ y \end{bmatrix}\]  (as A is of order 2)


How to Find Eigenvector

The following are the steps to find eigenvectors of a matrix:

Step 1: Determine 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​, …

Step 2: Substitute the value of λ1​ in equation AX = λ1​ X or (A – λ1​ I) X = O.

Step 3: Calculate the value of eigenvector X which is associated with eigenvalue λ1​.

Step 4: Repeat steps 3 and 4 for other eigenvalues λ2​, λ3​, … as well.


Generalized Eigenvector

Eigenvectors are not very different from generalized eigenvectors. It is defined in the following way:

A generalized eigenvector associated with an eigenvalue λ of an n times n×n matrix is denoted by a nonzero vector X and is defined as:

(A−λI)k = 0

Where k is some positive integer.

For k = 1 ⇒ (A−λI) = 0

Therefore, if k = 1, then eigenvector of matrix A is its generalized eigenvector.


Eigenvector Orthogonality

A vector quantity is known to possess magnitude as well as direction. Orthogonality is a concept of two eigenvectors of a matrix being at right angles to each other. We can say that when two eigenvectors are perpendicular to each other, they are said to be orthogonal eigenvectors.


Left Eigenvector

Eigenvector that is represented in the form of a row vector is called a left eigenvector. It satisfies the following condition:

AXL​=λXL

Where A is given matrix of order n and λ be one of its eigenvalues. XL​ is denoted by a row vector \[\begin{bmatrix} x_{1} & x_{2} & ... & x_{n} \end{bmatrix}\] 


Right Eigenvector

In the same way as the left eigenvector, the right eigenvector is denoted by XR​. It is defined as an eigenvector that is written in the form of a column vector, satisfying the condition given below:

AXR​=λXR

In which, A denotes an n×n square matrix and represents its eigenvalue.

X\[_{R}\] = \[\begin{bmatrix} x_{1}\\ x_{2}\\ . \\ . \\ . \\ x_{n} \end{bmatrix}\]


Power Method for Eigenvectors

Power method is another method for computing eigenvectors of a matrix. It is an iterative method that is used in numerical analysis. Power method works in the following way:


Let us assume that A be a matrix of order n×n and λ1​, λ2​,…,λn​ be its eigenvalues, such that λ1​ be the dominant eigenvalue. We are to select an initial approximate value x0​ for a dominant eigenvector of A.

Then

X\[_{1}\] = AX\[_{0}\] ……(1)

X\[_{2}\] = AX\[_{1}\] = AA(X\[_{0}\]) = A\[^{2}\] X\[_{0}\].....(using equation 1)

Similarly, we have

X\[_{3}\] = A\[^{3}\]X\[_{0}\]

X\[_{k}\] = A\[^{k}\]X\[_{0}\]


Solved Examples


1. Evaluate the Eigenvalues for the Following Matrix:

A = \[\begin{bmatrix} 4 & 6\\ 1 & 5 \end{bmatrix}\]


Solution:

Given,

A = \[\begin{bmatrix} 4 & 6\\ 1 & 5 \end{bmatrix}\]

Therefore,

A - \[\lambda\]I = \[\begin{bmatrix} 4 - \lambda & 6\\ 1 & 5 - \lambda \end{bmatrix}\] 

\[\mid\] A - \[\lambda\]I \[\mid\] = 0 

\[\mid\] A - \[\lambda\]I \[\mid\] = \[\begin{bmatrix} 4 - \lambda & 6\\ 1 & 5 - \lambda \end{bmatrix}\] = 0

\[\mid\] A - \[\lambda\]I \[\mid\] = (4 - \[\lambda\])(5 - \[\lambda\]) - 6 = 0

\[\mid\] A - \[\lambda\]I \[\mid\] = 20 - 5\[\lambda\] - 4\[\lambda\] + \[\lambda\]\[^{2}\] - 6 = 0

\[\mid\] A - \[\lambda\]I \[\mid\] = \[\lambda\]\[^{2}\] - 9\[\lambda\] + 14 = 0

\[\mid\] A - \[\lambda\]I \[\mid\] = (\[\lambda\] - 7)(\[\lambda\] - 2) = 0

\[\mid\] A - \[\lambda\]I \[\mid\] = \[\lambda\] = 7 or \[\lambda\] = 2


2. Evaluate the Eigenvectors for the Following Matrix:

A = \[\begin{bmatrix} 1 & 4\\ - 4 & -7 \end{bmatrix}\]


Solution:

A = \[\begin{bmatrix} 1 & 4\\ - 4 & -7 \end{bmatrix}\]

A - \[\lambda\]I =  \[\begin{bmatrix} 1-\lambda & 4\\ - 4 & -7 - \lambda \end{bmatrix}\]

 \[\mid\] A - \[\lambda\]I \[\mid\] = \[\begin{bmatrix} 1-\lambda & 4\\ - 4 & -7 - \lambda \end{bmatrix}\]

(1 - \[\lambda\])(- 7 - \[\lambda\])- 4 (-4) = 0

(\[\lambda\] + 3)\[^{2}\] = 0

\[\lambda\] = -3, -3

Using the eigenvector equation,

AX = -3X

A + 3I = 0

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

Which gives:

4x + 4y = 0

or

x + y = 0

Let us set x = k, then y = -k

Therefore, the required eigenvector is:

X = \[\begin{bmatrix} x \\ y \end{bmatrix}\] = k\[\begin{bmatrix} 1 \\ -1 \end{bmatrix}\] 

 

Did You Know

Eigenvectors are applicable in many fields in real life. Some of the important ones are illustrated below:

  1. Eigenvector decomposition is widely used in Mathematics in order to solve linear equations of the first order, in ranking matrices, in differential calculus etc.

  2. Eigenvectors are used in Physics to study simple modes of oscillation.

  3. This concept is widely used in Quantum Mechanics and Atomic and Molecular Physics. In the Hartree-Fock theory, the atomic and molecular orbitals are defined by the eigenvectors of the Fock operator.

  4. Eigenvectors are applied in almost all branches of engineering.

  5. Eigenvectors and Eigenvalues are used in Geology and the study of glacial till.

  6. The vibration analysis of mechanical structures with many degrees of freedom is done using eigenvalue problems. The eigenvalues denote the natural frequencies, also called the eigenfrequencies of vibration. The shapes of the vibration modes are denoted by the eigenvectors.

FAQs (Frequently Asked Questions)

Q1. Are All Eigenvectors Always Orthogonal?

For any matrix, the eigenvectors are not always orthogonal. However, in a symmetric matrix, where eigenvalues are always real, the corresponding eigenvalues are always orthogonal. A matrix A, multiplied with its transpose, yields a symmetric matrix in which the eigenvectors are always orthogonal. The principal component analysis is applied to the symmetric matrix, hence the eigenvectors will always be orthogonal.

Q2. Can a Real Matrix Have Complex Eigenvectors?

If an n×n matrix has all real values, then it is not necessary that the eigenvalues of the matrix are all real numbers. The eigenvalues are obtained from solutions of a quadratic polynomial. It is not always necessary for a quadratic polynomial to yield real values. For example: If a matrix has an eigenvalue like t2+1, then it will yield an imaginary result.