# Matrix A

Column
Row

## Introduction to the Eigenvectors Calculator

An eigenvector calculator is an online tool to evaluate eigenvalues and eigenvectors for a given matrix. It finds eigenvectors by finding the eigenvalues. It can evaluate the eigenvector corresponding to the eigenvalues.

In mathematics and data science, the concept of eigenvectors is most important because of its useful applications. It is used for different purposes, such as sometimes it is used to solve differential equations or may be used for linear transformation of a given matrix.

The concept of the eigenvector is tougher than the other concepts of matrices because it involves eigenvalues. You can get confused between eigenvectors and eigenvalues. To avoid this confusion, we introduced a tool online for students and mathematicians that can help find eigenvectors and provide an understanding of this concept.

## How to use Eigenvector of a Matrix Calculator

You can easily find the eigenvector for a given matrix using an eigenvector solver because it contains simple steps. These steps are:

1. In the first step, enter the value of the number of rows and columns in the respective boxes.
2. Now enter all values of all entries of the matrix.
3. You can also use the random option to select a random matrix.
4. Now click on the calculate button.

You will get the result a few seconds after clicking on the calculate button.

## How to Find an Eigenvectors Matrix Calculator?

You can find this tool from your browser by searching matrix eigenvectors calculator There will be a list of different math tools available on this site. Choose the eigenvector solver from the list.

## The Formula used by Eigenvector Calculator

The term eigenvector of a matrix refers to a vector associated with a set of linear equations. The linear transformation for the matrix A corresponding to the eigenvalue is given as:

$$Av \;=\; λv$$

Where,

v = Eigenvector of a given matrix A

λ = Eigenvalue of matrix A

The above equation can be rewrite to find eigenvector as:

$$(A \;-\; λI)v \;=\; 0$$

Where I is the identity matrix and 0 is the zero-vector. This formula is used by the eigenvector finder. Let’s see how an eigenvector can be found.

## How do you Find the Eigenvectors of a Matrix?

Suppose you have to find the eigenvector for matrix A which is given by:

$$A \;=\; \begin{bmatrix} 1 & 4 \\ -4 & -7 \\ \end{bmatrix}$$

Now,

$$|A \;-\; λI| \;=\; \begin{bmatrix} \begin{vmatrix} 1-λ & 4 \\ -4 & -7-λ \\ \end{vmatrix} \end{bmatrix}$$ $$(1 \;-\; λ)(-7 \; -λ) -4(-4) \;=\; 0$$ $$(λ \;+\; 3)^2 \;=\;0$$

Therefore,

$$λ \;=\; -3, \;-3$$

Now by using the eigenvalues in the eigen equation, the eigenvector can be calculated. So,

$$(A \;-\; λI)v \;=\; 0$$

Substituting the values of A, λ and I in the above equation.

$$\begin{pmatrix} \begin{bmatrix} 1 & 4 \\ -4 & -7 \\ \end{bmatrix} \;+\; λ \begin{bmatrix} 3 & 0 \\ 0 & 3 \\ \end{bmatrix} \end{pmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} \;=\; \begin{bmatrix} 0 \\ 0 \\ \end{bmatrix}$$ $$\begin{bmatrix} -λ+1 & 4 \\ -4 & -λ-7 \\ \end{bmatrix} v \;=\; 0$$

Substituting the value of λ.

$$\begin{bmatrix} 4 & 4 \\ -4 & -4 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} \;=\; \begin{bmatrix} 0 \\ 0 \\ \end{bmatrix}$$

We get,

$$x \;+\; y \;=\; 0$$

And,

$$x \;=\; -y$$

So, according to above equation, the eigenvector is:

$$v \;=\; \begin{bmatrix} -1 \\ 1 \\ \end{bmatrix}$$

## Why to use an Eigenvector Step-by-Step Calculator?

The concept of eigenvectors is important in mathematics because it is used to solve differential equations. You can make the process of finding eigenvectors easier with the use of this tool.

The use of an eigenvector finder tool is important because it will help you to understand the main concept by solving the problem step-by-step. That’s why you need to use this tool.

## Benefits of using Eigenvector Matrix Calculator

Since the internet has improved our lives, the learning and solving problems skills have also improved. This tool has many beneficial uses in improving your analytical skills to solve eigenvectors of 2x2, 3x3 and 4x4 matrices. These benefits are:

1. An eigenvector calculator can help you to find eigenvalues as well as eigenvectors easily.
2. It can save your time that can be consumed in manual computation.
3. It is easy to use because all you need to enter the matrix.
4. You can practice with different examples by using random options.
5. Eigenvectors calculator can help you to solve different applications related to data science and matrices. ### Hamza Haroon

Last Updated April 05, 2022

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