How to check if a vector is an eigenvector of a matrix?
It's very rigorous to use the definition of eigenvalue to know whether a scalar is an eigenvalue or not.
Answer: The product of an eigenvector with the matrix is always equal to the product of the eigenvector with the eigenvalue.
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
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