What is the characteristic equation?

Overview- characteristic equation

The characteristic equation is the equation which is used to find the Eigenvalues of a matrix.

This is also called the characteristic polynomial.

Definition- Let A be a square matrix, be any scalar then  is called the characteristic equation of a matrix A.

Note:

Let a be a square matrix and ‘ ’ be any scalar then,

1)     is called characteristic matrix

2)     is called characteristic polynomial.

The roots of a characteristic equation are known as characteristic root or latent roots, Eigenvalues or proper values of a matrix A.

Eigen vector

Suppose be an Eigenvalue of a matrix A. Then for every a non – zero vector x₁ such that.

                                                      … (1)

Such a vector ‘x₁’ is called an Eigenvector corresponding to the Eigenvalue .

Properties of Eigenvalues

1.     The sum of the Eigenvalues of a matrix A is equal to the sum of the diagonal elements of a matrix A.

2.     The product of all Eigenvalues of a matrix A is equal to the value of the determinant.

3.     If are n Eigenvalues of square matrix A then are m Eigenvalues of a matrix A^-1.

4.     The Eigenvalues of a symmetric matrix are all real.

5.      If all Eigenvalues are non –zero then A^-1 exist and conversely.

6.     The Eigenvalues of A and A’ are the same.

Properties of Eigenvector

1.     Eigenvectors corresponding to distinct Eigenvalues are linearly independent.

2.     If two more Eigen values are identical then the corresponding Eigenvectors may or may not be linearly independent.

3.     The Eigenvectors corresponding to distinct Eigenvalues of a real symmetric matrix are orthogonal.

Example: Find out the Eigenvalues and Eigenvectors of

Sol.  The Characteristics equation is given by

Hence the Eigen values are 0, 0 and 3.

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

This implies that x + y + z = 0

Here the number of unknowns is 3 and the number of equations is 1.

Hence we have (3-1) = 2 linearly independent solutions.

Let z = 0, y =  1 then x = -1 or let z = 1, y = 1 then x = -2

Thus the Eigenvectors corresponding to the Eigenvalue are (-1,1,0) and (-2,1,1).

The Eigenvector corresponding to Eigenvalue is

Where X is the column matrix of order 3, it means

This implies that -2x + y + z = 0

x – 2y + z = 0

x + y – 2z = 0

Taking last two equations we get

Or

Thus the Eigenvectors corresponding to the Eigenvalue are (3,3,3).

Hence the three Eigenvectors obtained are  (-1,1,0), (-2,1,1) and (3,3,3).

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