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Eigenvector calculator

This online calculator computes the eigenvectors of a square matrix up to the 4th degree.

This page exists due to the efforts of the following people:

Timur

Timur

Michele

Michele

Created: 2019-06-29 14:05:56, Last updated: 2020-12-11 12:18:53
Creative Commons Attribution/Share-Alike License 3.0 (Unported)

This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/8281/. Also, please do not modify any references to the original work (if any) contained in this content.

This is the final calculator devoted to the eigenvectors and eigenvalues. The first one was the Characteristic polynomial calculator, which produces a characteristic equation suitable for further processing. Second calculator - the Eigenvalue calculator solves that equation to find eigenvalues (using analytical methods, that's why it works only up to 4th degree), and the calculator below calculates eigenvectors for each eigenvalue found. Some theories can be found below the calculator.

PLANETCALC, Eigenvector calculator

Eigenvector calculator

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How to find Eigenvectors

Let me repeat the definition of eigenvectors and eigenvalues from the Eigenvalue calculator.

There are vectors for which matrix transformation produces the vector that is parallel to the original vector.

Av=\lambda v,

where \lambda is some scalar number.

These vectors are called the eigenvectors of A, and these numbers are called the eigenvalues of A.

We use the following form of the equation above: (A-\lambda I)v=0, where I is the identity matrix, to find the eigenvalues by solving the characteristic equation

det(A-\lambda I)=0.

After we found the eigenvalues, we can find eigenvectors. We should plug each concrete eigenvalue into the (A-\lambda I)v=0 equation and solve it for v. This means that we simply need to solve the following system of linear equations (in matrix form):

\begin{bmatrix}a_{11}-\lambda&a_{12}&\dots &a_{1n}\\a_{21}&a_{22}-\lambda&\dots &a_{2n}\\ \dots & \dots & \dots & \dots \\a_{n1}&a_{n2}&\dots &a_{nn}-\lambda\end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \\ ... \\ v_n\end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ ... \\ 0 \end{bmatrix}

This is a homogeneous system of linear equations, and even more, its equations are NOT independent. That is, the system has infinitely many solutions. This is because we have a family of eigenvectors (including zero vector), or eigenspace, for each eigenvalue. So, when you are asked to find eigenvectors for the matrix, you really need to pick up some "beautiful" solution for a system of linear equations obtained for each eigenvalue, that is, some sample eigenvector with possible no fractions and small positive integers.

In most cases, eigenvalue produces a homogeneous system with one independent variable. However, some cases have eigenvalue with multiplicity more than 1 (f.e. in case of double roots). In such cases, a homogeneous system will have more than one independent variable, and you will have several linearly independent eigenvectors associated with such eigenvalue - one for each independent variable.

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 #algebra #eigenvector Algebra characteristic equation eigenvalue eigenvector Math matrix
PLANETCALC, Eigenvector calculator
Timur2020-12-11 12:18:53

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