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 eigenvalues of a matrix.

10 Important Properties of Eigenvalues

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

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

Det (A) = λ1× λ2× …λn.

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

tr(A) = i=1nλi\sum_{i=1}^n \lambda_i

= λ1+ λ2+ …λn.

3. A matrix will have 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 | λi | = 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 square matrix and λ is an eigenvalue of A and n≥0 is an integer, then λn is an eigenvalue of An.

Example

Find the eigenvalues of A = [6345]\begin{bmatrix} -6 & -3\\ -4 & 5 \end{bmatrix}

Solution:

Given A = [6345]\begin{bmatrix} -6 & -3\\ -4 & 5 \end{bmatrix}

| A-λI | = 0

[6λ345λ]\begin{bmatrix} -6-\lambda & -3\\ -4 & 5-\lambda \end{bmatrix} = 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.