In linear algebra, an eigenvalue is a scalar value that represents the magnitude of a transformation of a linear transformation. An eigenvector is a non-zero vector that, when transformed by the linear transformation, only changes in magnitude (but not direction).

The eigenvalues and eigenvectors of a matrix are important because they provide information about the matrix and how it transforms vectors. For example, the eigenvalues of a matrix can be used to determine whether the matrix is diagonalizable, and the eigenvectors can be used to determine the matrix’s eigenspaces.

To find the eigenvalues and eigenvectors of a matrix, you need to solve the matrix’s characteristic equation. The characteristic equation is a polynomial equation in which the matrix is the variable and the eigenvalues are the roots. For example, if A is a 3×3 matrix, the characteristic equation is given by

|A – λI| = 0,

where λ is the eigenvalue and I is the identity matrix.

To find the eigenvectors of a matrix, you need to solve the system of equations given by

(A – λI)v = 0,

where v is the eigenvector.

We know that,

AX = λX

- AX – λX = 0
- (A – λI) X = 0… .. (1)

The above condition will be true only if (A – λI) is singular. That means,

| A-λI | = 0….. (2)

(2) is known as the characteristic equation of the matrix.

The roots of the characteristic equation are the eigenvalues of the matrix A.

Now, to find the eigenvectors, we simply put each eigenvalue into (1) and solve by Gaussian elimination, that is, we convert the augmented matrix (A – λI) = 0 to row-echelon form and solve the linear system of equations thus obtained.

- The eigenvalues of real symmetric and Hermitian arrays are real.
- The eigenvalues of real skew symmetric and Hermitian skew arrays are pure imaginary or zero.
- The eigenvalues of unitary and orthogonal arrays are of unitary modulus |λ| = 1.
- If λ
_{1}, λ_{2}…… .λ_{n}are the eigenvalues of A, then kλ_{1}, kλ_{2}…… .kλ_{n}are the eigenvalues of kA - If λ
_{1}, λ_{2}…… .λ_{n }are the eigenvalues of A, then 1/λ_{1}, 1/λ_{2}…… .1/λ_{n}are eigenvalues of A -1. - If λ 1, λ 2 …… .λ n are the eigenvalues of A, then λ 1 k , λ 2 k …… .λ n k are the eigenvalues of A k.
- Eigenvalues of A = Eigenvalues of A
^{T}(Transpose). - Sum of eigenvalues = Trace of A (Sum of diagonal elements of A).
- Product of eigenvalues = |A|.
- Maximum number of eigenvalues other than A = Size of A.
- If A and B are two arrays of the same order, then eigenvalues of AB = eigenvalues of BA.