Properties of Eigenvalues

A matrix is a rectangular arrangement of numbers in the form of rows and columns. Eigenvalues are a set of scalars related to the matrix equation. They are also known as characteristic roots or characteristic values. Consider an n×n matrix A. If AX = λA, then λ is the eigenvalue of the matrix A. X denotes the Eigen matrix of A. As far as the JEE exam is concerned, matrix is an important topic. In this article, we will learn the properties of the eigenvalues of a matrix.

10 Important Properties of Eigenvalues

Let A be a matrix with eigenvalues λ₁, λ₂, …λ_n.

1. The determinant of A is the product of all the eigenvalues of A.

Det (A) = λ₁ × λ₂ × …λ_n.

2. The trace of A is the sum of all the eigenvalues of A.

= λ₁+ λ₂+ …λ_n.

3. A matrix will have an inverse if and only if all of its eigenvalues are nonzero.

4. Eigenvalue can be Zero

5. If A is an n × n triangular matrix (upper triangular, lower triangular, or diagonal), then the eigenvalues of A are the entries of the main diagonal of A.

6. If an n × n matrix A has n distinct eigenvalues, then A is a diagonalizable matrix.

7. If A is unitary, every eigenvalue has absolute value | λ_i | = 1.

8. If A is Hermitian (symmetric) matrix, then the eigenvalues of A are all real numbers.

9. If A is a square matrix, then for every eigenvalue of A, the geometric multiplicity is less than or equal to the algebraic multiplicity.

10. If A is a square matrix and λ is an eigenvalue of A, and n ≥ 0 is an integer, then λⁿ is an eigenvalue of Aⁿ.

Example

Solution:

Given,

| A – λI | = 0

(-6 – λ)(5 – λ) – 12 = 0

-30 – 5λ + 6λ + λ² – 12 = 0

λ² + λ – 42 = 0

(λ + 7)(λ – 6)= 0

λ = -7 or λ = 6

Hence, the required eigenvalues are -7 and 6.

Video Lessons

Matrices and Determinants – Important Topics

Matrices and Determinants – Important Questions