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Eigenvectors of a Matrix

Tool to calculate eigenvectors of a matrix. Eigenvectors of a matrix are vectors whose direction remains unchanged after multiplying by the matrix. They are associated with an eigenvalue.

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Eigenvectors of a Matrix -

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1. Mathematics
2. Matrix
3. Eigenvectors of a Matrix

Eigenvectors Calculator

Eigenspaces Calculator

Answers to Questions (FAQ)

What are eigen vectors of a matrix? (Definition)

An eigenvector of a matrix is a characteristic vector (or privileged axis or direction) on which a linear transformation behaves like a scalar multiplication by a constant named eigenvalue.

The set of eigenvectors form an eigenspace.

How to calculate eigenvectors of a matrix?

To find eigenvectors, take $ M $ a square matrix of size $ n $ and $ \lambda_i $ its eigenvalues. Eigenvectors are the solution of the system $ ( M − \lambda I_n ) \vec{X} = \vec{0} $ with $ I_n $ the identity matrix.

Example: The 2x2 matrix $$ M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} $$
Eigenvalues for the matrix $ M $ are $ \lambda_1 = 5 $ and $ \lambda_2 = -1 $ (see tool for calculating matrix eigenvalues).

For each eigenvalue, look for the associated eigenvector.

Example: For $ \lambda_1 = 5 $, solve $ ( M − 5 I_n ) X = \vec{0} $: $$ \begin{bmatrix} 1-5 & 2 \\ 4 & 3-5 \end{bmatrix} . \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$ and find as solution $$ \begin{align} -4 x_1 + 2 x_2 &= 0 \\ 4 x_1 - 2 x_2 &= 0 \end{align} \iff \begin{array}{c} x_1 = 1 \\ x_2 = 2 \end{array} $$ So the eigenvector associated to $ \lambda_1 = 5 $ is $ \begin{pmatrix} 1 \\ 2 \end{pmatrix} $

Example: For $ \lambda_2 = -1 $, solve $ ( M + I_n ) X = \vec{0} $ like this: $$ \begin{bmatrix} 1+1 & 2 \\ 4 & 3+1 \end{bmatrix} . \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \\ \begin{align} 2 x_1 + 2 x_2 &= 0 \\ 4 x_1 + 4 x_2 &= 0 \end{align} \iff \begin{array}{c} x_1 = -1 \\ x_2 = 1 \end{array} $$

So the eigenvector associated to $ \lambda_1 = -1 $ is $ \begin{pmatrix} -1 \\ 1 \end{pmatrix} $.

How to prove that a matrix is diagonalizable?

Does a zero vector as an eigenvector exist?

The definition of the eigenvector precludes its nullity. However, if in a calculation the number of independent eigenvectors is less than the number of eigenvalues, dCode will sometimes display a null vector.

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This calculator allows you to enter any square matrix from 2x2, 3x3, 4x4 all the way up to 9x9 size. It will find the eigenvalues of that matrix, and also outputs the corresponding eigenvectors.

For background on these concepts, see 7. Eigenvalues and Eigenvectors

Instructions

First, choose the matrix size you want to enter. You will see a randomly generated matrix to give you an idea of what your output will look like.

Then, enter your own numbers in the boxes that appear. You can enter integers or decimals. (More advanced entry and output is in the works, but not available yet.)

On a keyboard, you can use the tab key to easily move to the next matrix entry box.

Click calculate when ready.

The output will involve either real and/or complex eigenvalues and eigenvector entries.

You can change the precision (number of significant digits) of the answers, using the pull-down menu.

Eigenvalues and eigenvectors calculator

Matrix size: Precision:

NOTE 1: The eigenvector output you see here may not be the same as what you obtain on paper. Remember, you can have any scalar multiple of the eigenvector, and it will still be an eigenvector. The convention used here is eigenvectors have been scaled so the final entry is 1.

NOTE 2: The larger matrices involve a lot of calculation, so expect the answer to take a bit longer.

NOTE 3: Eigenvectors are usually column vectors, but the larger ones would take up a lot of vertical space, so they are written horizontally, with a "T" superscript (known as the transpose of the matrix).

NOTE 4: When there are complex eigenvalues, there's always an even number of them, and they always appear as a complex conjugate pair, e.g. 3 + 5i and 3 − 5i.

NOTE 5: When there are eigenvectors with complex elements, there's always an even number of such eigenvectors, and the corresponding elements always appear as complex conjugate pairs. (It may take some manipulating by multiplying each element by a complex number to see this is so in some cases.)

Credit: This calculator was built using the Numeric.js library.

How many eigenvectors does a 2x2 matrix have?

How do you find eigenvectors of a 2x2 matrix with eigenvalues?