Eigenvalues: Overview, Questions, Preparation

Eigenvalues consist of a set of scalar values that are related to a linear equation. You can use it to solve matrix equations. Eigen means characteristic in German, and therefore Eigenvalues means characteristic root or proper value. The equation that is used to represent eigenvalues is given as below:

AX = λX

λ is the scalar value here. 

Why are Eigenvalues Used?

Eigenvalues are used to transform a given vector. It is the scalar that points in the direction in which the eigenvector transforms. This means that the eigenvector’s direction of transformation is positive if the eigenvalue is positive and a negative eigenvalue denotes that the eigenvector’s direction of transformation is negative. 

What is an Eigenvector?

An eigenvector is a vector that remains unchanged in a direction even when it is subjected to a linear transformation. It changes its direction only when a scalar value is applied to it. Suppose that the linear transformation of a vector that falls in space V is denoted by A and its vector is X then the eigenvector v has a scalar multiple A(X). 

Eigenvalues: Properties

Two eigenvectors are independent (linearly) if they have separate eigenvalues. 

The singular matrices do not have any eigenvalue.

If λ = 0 then it cannot be called as an eigenvalue of a square matrix A. 

Matrix scalar multiple: If λ is an eigenvalue of a square matrix A then aλ is said to be aA’s eigenvalue. 

Power of matrix: If λ is an eigenvalue of a square matrix A then λn is said to be An eigenvalue.

The eigenvalue of an inverse matrix: If λ is an eigenvalue of a square matrix A then A-1 has an eigenvalue λ-1

The eigenvalue of a transpose matrix: If λ is an eigenvalue of a square matrix A, transpose matrix At  has an eigenvalue λ.

A matrix’s polynomial: If p(x) is a polynomial of a square matrix A that has an eigenvalue λ, then matrix p(A) will have the eigenvalue p(λ).

Eigenvalues topic is introduced in earlier classes, but it is also covered in Class XII’s chapter of Determinants. It has a weightage of 2 to 3 marks depending on the question paper set on that particular year. 

1. Find the eigenvalues of the matrix A = |4    1|
                                                                      |1    4|

Solution.

The below formula can calculate the determinant of a characteristic equation:

det (A - λI) = det |4 - λ     1|
                          |1     4 - λ|

                     = (4 - λ) (4  λ ) - 1 =  λ2- 8λ  + 15 = (λ - 5) (λ - 3) = 0 

Therefore, the matrix A has two eigenvalues 5 and 3.

2. Find the eigenvalues of the matrix B = |1    0|
                                                                      |1    3|

Solution.

We have the formula det (A - λI) = 0 

Therefore, |1    0| - λ |1    0| = 0
                     |1    3|       |0    1|

|1 - λ    0|  = 0 
|0    3 - λ|

By solving this, we get two eigenvalues, λ1 = 3 and λ2 = 1.

Q: What do you mean by eigenroots?

Q: One eigenvalue can be related to how many eigenvectors?

Q: Why are eigenvalues used?

Q: In which field is the eigenvalue used extensively?

Q: Can an eigenvalue have a negative value?