Eigenvector: Overview, Questions, Preparation

A vector that is associated with a set of linear equations is the eigenvector. It is also known as a latent vector, proper vector, or characteristic vector. These are specified in a square matrix reference. Eigenvectors and many other applications related to them are also useful in solving differential equations.

Source: NCERT

Definition of Eigenvector 

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

Av = λv 

Where the 

V = Eigenvector, and λ is the scalar quantity associated with the matrix A given as its own value. 

Equation of Eigenvector 

The equation corresponding to each matrix eigenvalue is given by: 

AX = λ X   

It is known formally as the equation of the eigenvector. 

Substitute each eigenvalue in place of λ and get the eigenvector equation that allows us to solve each eigenvector belonging to each eigenvalue. 

Method of Determination of the Eigenvector 

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

1. If A is a matrix of n and λ is the value associated with it. The following relationship can then be defined by the eigenvector v: Av = λvv 
2. If "I" is the identity matrix of the same order as A, then the identity matrix is the same as A (A – λI)v = 0 
3. It is possible to determine the eigenvector associated with matrix A using the above method. 

Follow the procedure given below to find the eigenvectors of a matrix: 

1. Using the equation det ((A-λI) =0, where "I" is equal to the order identity matrix as A, find the eigenvalues of the given matrix A. Denote each of the λ1, λ2, λ3 eigenvalues. 
2. The values in the equation AX = λ1 or (A- λ1 I) X = 0 are replaced. 
3. Calculate the value of the eigenvector X associated with the value of the eigenvector. 
4. For the remaining eigenvalues, repeat the steps to find the eigenvector.

Applications of Eigenvector 

The significant application of eigenvectors is as follows: 

1. In physics, eigenvectors are used in simple oscillation mode. 
2. Eigenvector Decomposition is widely used in mathematics to solve first-order linear equations, ranking matrices, differential calculus, etc. 
3. In quantum mechanics, this concept is widely used. 
4. It is applicable in almost all engineering sectors.

This topic is taught under the chapter Determinants. You will learn about the eigenvectors, their conditions and use them to solve determinant questions. The weightage of this chapter is 8 marks in the final exam.

1.Estimate the eigenvalues of a matrix.

Solution:
The eigenvalues of the given matrix are 2 and 5.

2. Estimate the eigenvalues of a matrix.

Solution:
The eigenvalues of the given matrix are 3, 4, and 7.

3. Estimate the eigenvalues of a matrix.

Solution:
The eigenvalues of the given matrix are -2 and 6.

Q: What is a matrix eigenvector? 

Q: What is the usage of the eigenvalues and the eigenvectors? 

Q: What is the PCA eigenvector? 

Q: What do eigenvalues inform us? 

Q: Can an eigenvalue of its own have no eigenvector?