# Eigenvector calculator | Eigenvalue calculator

## Eigenvalue and eigenvector calculator

Eigenvalue and eigenvector calculator allows you to calculate the eigenvalues and eigenvectors of any square matrix quickly and easily. To use it, you only need to enter the values of the matrix and press the “calculate” button. When doing so, the values and eigenvectors of the entered matrix will automatically be displayed.

## Eigenvalues and eigenvectors

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.

## How to find Eigenvectors and Eigenvalues?

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.

## Eigenvalue Properties

• 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 λ1, λ2 …… .λn are the eigenvalues ​​of A, then kλ1 , kλ2 …… .kλn are the eigenvalues ​​of kA
• If λ1, λ2 …… .λn are the eigenvalues ​​of A, then 1/λ1 , 1/λ2 …… .1/λn 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 AT (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.