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.
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.
- 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.
Let us take an example to understand the eigenvalues concept.
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