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**Table of Content**

Eigenvector calculator is use to calculate the eigenvectors, multiplicity, and roots of the given square matrix.

This calculator also finds the eigenspace that is associated with each characteristic polynomial. In this context, you can understand how to find eigenvectors 3 x 3 and 2 x 2 matrixes with the eigenvector equation.

In linear algebra, an eigenvector of a linear transformation is a non-zero vector that changes by a scalar factor when that linear transformation is applied to it. The corresponding value, often denoted by λ, is the factor by which the eigenvector is scaled.

Let’s consider that A is an n x n square matrix, and if v is a non-zero vector, then we can say that the product of matrix A and vector v is the product of a scalar quantity λ and the given vector, such that:

Av =λv

Where

v = Eigenvector

λ be the scalar quantity that is known as the eigenvalue associated with the given matrix A

The technique of determining the eigenvector of a matrix/ linear equation is given as follows:

If A is an n×n matrix and λ is the eigenvalues related to it. Then, eigenvector v can be described in the following respect:

Av =λv

If “I” be the identity matrix of the identical order as A, then

(A – λI)v =0

The eigenvector corresponding with matrix A can be estimated using the above method.

Here, “v” is termed as the eigenvector belonging to each eigenvalue and is expressed as:

$$ \begin{array}{l}v =\begin{bmatrix} v_{1}\\ v_{2}\\ .\\ .\\ v_{n}\end{bmatrix}\end{array} $$

However, our online generalized eigenvector calculator is an easy way to perform calculations.

The basic representation of the relationship between a eigenvector and its corresponding value is

$$ Xv = λv $$

Where

- v is a vector with m columns
- A is a matrix with m rows and m columns
- λ is a scalar.

In this relation, the true value of v is the eigenvector. In order for the variable to be true, it must satisfy the equation so that the left side and the right side of the equation are the same.

The eigenvector satisfies the equation for any given eigenvalue. There can be more eigenvectors than eigenvalues, so each λ value can have multiple v values that satisfy the equation. The value can have an infinite number of eigenvectors, but there are usually only a few different eigenvectors.

Xv = λv can be converted to A – I = 0, where I is the identity matrix. Then you can start multiplying and subtracting matrices to get polynomials. If the eigenvalues are known, then we can insert them into the equation Xv = λv and find our vector.

The basis for the eigenvalue calculator with steps computes the eigenvector of given matrixes quickly by following these instructions:

- Select the size of the matrix (such as 2 x 2 or 3 x 3) from the drop-down list of the eigenvector finder.
- Insert the values into the relevant boxes eigenvector solver.
- You can generate a matrix by clicking on generate matrix button.
- If you need to remove values, then click on clear all fields.
- Click on the calculate button.

- The eigenvalue and matrix eigenvector calculator provides the multiplicity, Eigen vectors, and values of the given matrix.
- This find eigenvectors calculator takes the determinant of the obtained matrix and solves the equation for getting the roots.
- It eigenvector matrix calculator computes eigenvectors and displays a step-by-step calculation for eigenvectors.

From the source of Wikipedia: Eigensystem, eigen vectors of matrices, characteristic polynomial, Diagonal matrix, Eigenfunctions of differential operators.