Eigenvalues and Eigenvectors Calculator

Eigenvalues and Eigenvectors Calculator

This calculator is used to calculate eigenvalues and eigenvectors (eigenspace) of a given square matrix and shows step-by-step solutions. Our online eigenvalue eigenvector calculator allows you to enter any square matrix from 2×2, 3×3, 4×4, and 5×5 sizes. In this article, we will try to discuss:

⁍ What is an Eigenvalues & Eigenvectors;
⁍ Formula to Find Eigenvalue & Eigenvector;
⁍ Relation between Eigenvalue & Eigenvector;
⁍ How to Calculate Eigenpair with a Step-by-Step Example;
⁍ How to Use this Eigenvalue and Eigenvector Calculator;
⁍ Eigenvalue and Eigenvector Decomposition of a Matrix;
⁍ What is the Importance of Eigenvalues and Eigenvectors;

What is an Eigenvalues & Eigenvectors?

Definition of Eigenvalue:

An eigenvalue is a scalar (number) λ associated with a square matrix A. It describes how a linear transformation changes a corresponding eigenvector, which is a non-zero vector that only gets scaled, not changed in direction. 

Definition of Eigenvector:

An eigenvector of a matrix is a non-zero vector that, when a linear transformation is applied to it, changes only in scale but does not change its direction. In the equation Av = λv, where A is a matrix and v is an eigenvector. 

Formula to Calculate Eigenvalue & Eigenvector:

➦ Eigenvalues

Characteristic equation for the eigenvalues is given:

det( A − λI ) = 0

➦ Eigenvectors

The vector equation for an eigenvector (eigenspace) is:

( A − λI )v = 0

In both formulas, where:

• A → square matrix
• λ → eigenvalue
• I → identity matrix
• v → eigenvector

Relation between Eigenvalue & Eigenvector:

The fundamental relationship between eigenvalues and eigenvectors with respect to the matrix A is:

(Av = λv)

This equation means that when the matrix 'A' is applied to a special non-zero vector 'v', the result is only a scaling of that vector by 'λ', without changing its direction (and it flips direction if λ is negative).

The eigenvalue represents the scale of transformation, while the eigenvector is the special, non-zero vector that gets scaled. To express scalar multiplication as a special case of matrix multiplication, λv can be written as λIv, where I is the identity matrix.

Rearranging the terms gives the standard eigenvalue equation:

( A − λI ) v = 0

This equation is centered on determining the eigenvalues and eigenvectors of a matrix.

? The determinant of a matrix is helpful in many mathematical operations. Suppose it is used to check whether a matrix has an inverse and to calculate that inverse using our inverse matrix calculator.

How to Calculate the Eigenvalue & Eigenvector?

Calculating eigenvalues and eigenvectors step by step becomes easy when explained through 2x2 matrix examples. Using our eigenvalues and eigenvectors calculator helps you to solve 2×2, 3×3, 4×4, and 5×5 sizes matrix calculations within simple steps. But here we try to solve a 2x2 matrix example and solve it step-by-step, which demonstrates each step to know how to find eigenvalues and eigenvectors.

Let the matrix be:

\(A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \)

Find the Eigenvalues:

3.1 Create the characteristic equation

det ( A − λI ) = 0

Where I is the identity matrix:

\( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)

So, 

\( A - \lambda I =\begin{bmatrix}2 - \lambda & 1 \\1 & 2 - \lambda\end{bmatrix}\)

3.2 Take the determinant

( 2 − λ ) ( 2 − λ ) − ( 1 ) ( 1 ) = 0

( 2 − λ )² − ( 1 ) = 0 

3.3 Solve the equation

(2−λ)² = 1

2 − λ = ±1

So the eigenvalues are:

λ₁ = 3, λ₂ = 1

Find Eigenvectors (one by one)

Eigenvector for λ = 3

4.1 Put λ = 3 into A − λI

\( A - I = \begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix} \)

4.2 Solve ( A − λI ) v = 0

− x + y = 0

y = x

4.3 Write the eigenvector

\( V₁ = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \)

Eigenvector for λ = 1

4.4 Put λ = 1 into A − λI

\( V₂ = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \)

4.5 Solve the equation

x + y = 0

y = −x

4.6 Write the eigenvector

\( A - I =\begin{bmatrix} 1 & 1 \\1 & 1\end{bmatrix} \)

? Matrix multiplication is another important operation closely related to matrices. It is often used alongside eigenvalues and eigenvectors in solving linear algebra problems, and you can easily perform these calculations using our matrix multiplication calculator.

How to Use this Eigenvalue and Eigenvector Calculator?

1. Choose the matrix size 2x2, 3x3, 4x4, or 5x5 from the respective input field
2. Then, enter the numbers in the boxes accordingly (you can enter integers or decimals)
3. Click on the "CALCULATE" button to get an instant and accurate answer

Eigenvalue and Eigenvector Decomposition of a Matrix:

Eigenvalue and eigenvector decomposition show a matrix A using its eigenvalues and eigenvectors:

A = PDP⁻¹

• P → Matrix of eigenvectors
• D → Diagonal matrix of eigenvalues
• P⁻¹ → Inverse of matrix P

Important Notes:

✔️ Eigenvalue decomposition can be done only if matrix A is diagonalizable, which has enough independent eigenvectors to form a full set.

✔️ The order of eigenvalues in D should be in the same order as eigenvectors in P.

✔️ If A is a symmetric matrix, then P⁻¹ = Pᵀ and the decomposition becomes:

A = PDPᵀ

What is the Importance of Eigenvalues and Eigenvectors?

• System Dynamics: Analyzing stability in differential equations, including whether the systems are stable, unstable, or oscillatory.
• Quantum Mechanics: Eigenvalues and eigenvectors describe states and correspond to measurable quantities.
• Principal Component Analysis (PCA): Reducing dimensions in data science and uses eigenvectors of the covariance matrix to find principal directions.
• Vibrations and Mechanical Systems: Eigenvalues help determine natural frequencies in structures and machines.
• Markov Chains & Probability: Eigenvectors/eigenvalues help to find steady state distributions in stochastic processes.
• Diagonalization: It allows us to simplify matrices to diagonal form, which simplifies many calculations.
• Graph Theory & Networks: Eigenvalues of adjacency or Laplacian matrices reveal connectivity and network properties.

FAQs

What does it mean when a matrix has multiple eigenvalues?

When a matrix has several eigenvalues, it implies the matrix acts on vectors in multiple distinct ways. 

• A matrix may possess repeated eigenvalues (occurring more than once) or distinct eigenvalues.
• When an eigenvalue is repeated, it might have one or more corresponding eigenvectors.

Can this calculator handle 3×3 matrices?

Yes, this matrix eigenvalues and eigenvectors calculator can handle matrices in 2×2, 3×3, 4×4, and 5×5 sizes.

How to Find Eigenvectors Once Eigenvalues Are Known?

Once the eigenvalues are known, then the eigenvectors can be found by using systems of linear equations:

Steps:

1. Take the equation ( A − λI ) v = 0 
2. Put a known eigenvalue λ. 
3. Solve the resulting system for vector v.
4. Any non-zero solution of this system is an eigenvector.
5. Repeat these steps for each eigenvalue.

Are there any other names of eigenvalue?

The eigenvalue is the most commonly used term, the others are considered older or alternative terminology.

• latent roots
• characteristic root
• characteristic value
• the proper value

References:

Wikipedia: Eigenvalues and eigenvectors, Spectrum of a matrix, Algebraic multiplicity, Eigenspaces, geometric multiplicity, and the eigenbasis for matrices, Left and right eigenvectors, Diagonalization and the eigendecomposition.

Libretext Mathematics: Eigenvalues and Eigenvectors of a Matrix, Finding Eigenvectors and Eigenvalues, Eigenvalues and Eigenvectors for Special Types of Matrices

Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 145, 1988.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Eigensystems." Ch. 11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 449-489, 1992.