# Eigenvalues

Eigenvalues are associated with eigenvectors in Linear algebra. Both terms are used in the analysis of linear transformations. ‘Eigen’ is a German word which means ‘proper’ or ‘characteristic’. Therefore, the term eigenvalue can be termed as characteristics value or characteristics root as well.

Eigenvectors are the vectors (non-zero) which do not change the direction when any linear transformation is applied. It changes by only a scalar factor. In a brief, we can say, if A is a linear transformation from a vector space V and X is a vector in V, which is not a zero vector, then v is an eigenvector of A if A(X) is a scalar multiple of X. We can write this condition as;

AX = λX

An eigenspace of vector X consists of set of all eigenvectors with the equivalent eigenvalue collectively with the zero vector. Though, the zero vector is not an eigenvector.

## Eigenvectors

Let us say A is a n × n matrix and λ is an eigenvalue of matrix A, then X, a non-zero vector, is called as eigenvector, if it satisfies the given below expression;

AX = λX

X is an eigenvector of A corresponding to eigenvalue, λ.

Note:

• There could be infinitely many Eigenvectors, corresponding to one eigenvalue.
• For distinct eigenvalues, the eigenvectors are linearly dependent.

## Eigen matrix of a square matrix

Suppose, An×n is a square matrix, then [A- λI] is called an eigen or characteristic matrix, which is an indefinite or undefined scalar. Where determinant of eigen matrix can be written as, |A- λI| and |A- λI| = 0 is the eigen equation or characteristics equation, where I is the identity matrix. The roots of an eigen matrix are called eigen roots.

Eigenvalues of a triangular matrix and diagonal matrix are equivalent to the elements on the principal diagonals. But eigenvalues of the scalar matrix are the scalar only.

### Eigenvalues Example

Let us take an example to understand the eigenvalues concept.