# Eigenvalues and Eigenvectors Calculator

## Calculate eigenvalues and eigenvectors step by step

The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown.

Related calculator: Characteristic Polynomial Calculator

This Eigenvalue and Eigenvector Calculator is an advanced tool designed to provide you with precise and quick calculations of eigenvalues and eigenvectors. It works with any square matrix and handles complex eigenvalues with absolute accuracy. Calculating eigenvalues and corresponding eigenvectors of a matrix has never been easier.

## How to Use the Eigenvalues and Eigenvectors Calculator?

### Input the Square Matrix

The first step is to enter your matrix values. Make sure your matrix is a square matrix, which means it must have the same number of rows and columns.

### Enter the Values

In the provided grid, input the values of your square matrix. You should fill out each cell corresponding to the matrix's row and column positions.

### Click "Calculate"

Once your matrix values have been entered, click on the "Calculate" button.

### Review the Results

After clicking the "Calculate" button, the calculator will compute the eigenvalues and eigenvectors of the input matrix and display the results. These will include all the eigenvalues and their corresponding eigenvectors.

## Understanding Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors hold an integral place in linear algebra. They serve as crucial elements in a range of mathematical operations and find use in diverse disciplines such as physics, engineering, and computer science, among others.

Consider a square matrix $$$A$$$. If there is a nonzero vector $$$\mathbf{\vec{v}}$$$ that, when multiplied by $$$A$$$, results in a vector which is a scaled version of $$$\mathbf{\vec{v}}$$$ (let $$$\lambda$$$ is a scale factor), then $$$A$$$ is said to have an eigenvalue $$$\lambda$$$. This scaled value is referred to as the eigenvalue, and the vector $$$\mathbf{\vec{v}}$$$ that brings about this scalar multiplication is recognized as the corresponding eigenvector.

In formulaic terms:

$$A\mathbf{\vec{v}}=\lambda\mathbf{\vec{v}}$$where

$$$A$$$ = square matrix,

$$$\mathbf{\vec{v}}$$$ = eigenvector,

$$$\lambda$$$ = eigenvalue.

## Characteristic Polynomial and Identity Matrix

The determination of a matrix's eigenvalues necessitates the resolution of what is known as the characteristic polynomial. This polynomial arises from subtracting the eigenvalue $$$\lambda$$$ multiplied by an identity matrix $$$I$$$ (of the same size as $$$A$$$) from the original matrix $$$A$$$.

The characteristic equation, symbolized as $$$\left|A-\lambda I\right|=0$$$ (where $$$\left|A-\lambda I\right|$$$ is the determinant of a matrix $$$A-\lambda I$$$), yields the roots which are the matrix's eigenvalues.

## Complex Eigenvalues and Eigenvectors

Our calculator for eigenvectors is fully capable of dealing with complex eigenvalues. Complex eigenvalues arise when the solutions of the characteristic equation manifest as complex numbers. Accompanying these complex eigenvalues are their respective complex eigenvectors.

## Why Choose Our Eigenvalues and Eigenvectors Calculator?

### Precision and Speed

Our calculator provides you with accurate results quickly, saving you from time-consuming manual calculations and reducing the possibility of errors.

### Handles Complex Eigenvalues

Unlike some calculators, ours is capable of managing complex eigenvalues, ensuring that you can handle all types of problems.

### Versatility

It can work with many sizes of a square matrix. Whether your matrix is 2x2 or 4x4, our calculator can handle it.

### Educational Value

By seeing how the calculator works, students can better understand the process of calculating eigenvalues and eigenvectors, enhancing their learning experience.

### FAQ

#### What is an eigenvalue?

In linear algebra, an eigenvalue is a scalar associated with a given linear transformation that characterizes the way the transformation scales a given vector.

#### What is an eigenvector?

An eigenvector is a nonzero vector that is only changed by a scalar factor when a linear transformation is applied to it.

#### Why am I getting a complex number as my eigenvalue?

A complex eigenvalue results when the roots of the characteristic equation are complex numbers. This is common and expected in certain types of matrices.

#### How is the characteristic polynomial used to find eigenvalues?

The characteristic polynomial arises from the determinant of the difference between the matrix and the product of the eigenvalue and an identity matrix. The roots of this polynomial equation are the eigenvalues of the matrix.