How to check if a vector is an eigenvector of a matrix?

Solution:

It's very rigorous to use the definition of eigenvalue to know whether a scalar is an eigenvalue or not.

It is an easy practice to find the roots of the characteristic polynomial.

Eigenvectors are defined by the equation:

| A - λI | = 0

Ax = 𝜆x = 𝜆Ix

A is the matrix whose eigenvector is been checked,

where 𝜆 = eigenvector, I = unit matrix.

From the above equation, on further simplification we get:

⇒ (A − 𝜆I) x = 0  ( taking x as common )

⇒ | A - λI | = 0 

Thus indicating that A = 𝜆I

Therefore, an eigenvector is a vector which when multiplied with the matrix gives the same result as that of the product of an eigenvalue with the eigenvector as mentioned in the above proof of the eigenvector x.

How to check if a vector is an eigenvector of a matrix?

Summary:

The product of an eigenvector with the matrix is always equal to the product of the eigenvector with the eigenvalue.