 Eigenvalues

Eigenvalues are associated with eigenvectors in Linear algebra. Both terms are used in the analysis of linear transformations. Let us discuss in the definition of eigenvalue, eigenvectors with examples in this article.

Eigenvalue Definition

Eigenvalues are the special set of scalars associated with the system of linear equations. It is mostly used in matrix equations. ‘Eigen’ is a German word which means ‘proper’ or ‘characteristic’. Therefore, the term eigenvalue can be termed as characteristics value, characteristics root, proper values or latent roots as well. In simple words, the eigenvalue is a scalar that is used to transform the eigenvector. The basic equation is

AX = λX

The number or scalar value “λ” is an eigenvalue of A.

What are EigenVectors?

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.

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

Let us say A is an “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 Equation 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. 