What if an eigenvalue is repeated?

A repeated eigenvalue has algebraic multiplicity greater than one. Solve (A - λI)v = 0 and count the independent solutions: that is its geometric multiplicity. If they match, the eigenspace supplies enough eigenvectors; if there are fewer, the eigenvalue is defective and the matrix cannot be diagonalized. For [[2,1,0],[0,2,0],[0,0,3]], λ = 2 appears twice but has only one eigenvector, so the matrix is not diagonalizable. The calculator reports both multiplicities and, when possible, P and D with A = PDP^-1.

Eigenvalue and Eigenvector Calculator

Find a matrix’s invariant directions with MiniWebtool’s Eigenvalue and Eigenvector Calculator. A 2×2 or 3×3 matrix yields eigenvalues and associated eigenvectors, with characteristic-polynomial steps and transformation geometry.

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Help me understand MiniWebtool's Eigenvalue and Eigenvector Calculator: https://miniwebtool.com/eigenvalue-eigenvector-calculator/. What can I use it for, what inputs or settings do I need, and how do I use it and understand the output? Walk me through one simple example. Use this page as your source and answer in English.

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Eigenvectors keep their direction and are only stretched: Av = λv

Enter a 2×2 or 3×3 matrix below to get its eigenvalues and eigenvectors, the characteristic polynomial and the solution steps.

Try Example Matrices

Symmetric 2×2 Simple 2×2 Rotation 3×3 Example Diagonal 3×3 Repeated λ (defective) 4×4 Example

Matrix Size:

[

a₁₁

a₁₂

a₁₃

a₂₁

a₂₂

a₂₃

a₃₁

a₃₂

a₃₃

]

Or paste a matrix up to 6×6, one row per line

Spaces or commas between numbers; fractions like 1/2 work. When this box holds a matrix, the grid above is ignored.

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About Eigenvalue and Eigenvector Calculator

Welcome to the Eigenvalue and Eigenvector Calculator, a comprehensive tool for computing eigenvalues and eigenvectors of 2×2 and 3×3 matrices. This calculator provides detailed step-by-step solutions, derives the characteristic polynomial, analyzes matrix properties, and visualizes the transformation geometry. Ideal for students, teachers, engineers, and researchers working with linear algebra.

At a Glance

The Eigenvalue and Eigenvector Calculator finds the eigenvalues and eigenvectors of a 2x2 or 3x3 matrix. It shows the characteristic polynomial, step-by-step working and a picture of the matrix transformation.

Eigenvalue and Eigenvector Calculator: features at a glance
Feature	What you get	Good for
Matrix size	Choose a 2x2 or 3x3 matrix.	Linear algebra homework and small systems
Entry formats	Matrix entries can be integers, decimals or fractions such as 1/2.	Exact fractional matrices from a textbook
Characteristic polynomial	The polynomial det(A - lambda I) is derived, and its roots give the eigenvalues.	Following the method rather than only the answer
Matrix properties	Reports the trace, determinant, symmetry and definiteness of the matrix.	Checking that eigenvalues sum to the trace
Visualization	Eigenvectors and the matrix transformation are drawn as a picture.	Seeing which directions the matrix only scales
Step-by-step solution	Each stage is explained from the characteristic equation to the eigenvectors.	Students learning the procedure

What Are Eigenvalues and Eigenvectors?

In linear algebra, eigenvalues and eigenvectors are fundamental properties of square matrices that reveal how the matrix transforms vectors. An eigenvector is a non-zero vector that, when the matrix acts on it, only changes in scale (not direction). The scaling factor is the corresponding eigenvalue.

Eigenvalue-Eigenvector Equation

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

Where:

A is a square matrix (n×n)
v is an eigenvector (non-zero vector)
λ (lambda) is the eigenvalue (scalar)

Geometrically, eigenvectors point in directions that remain unchanged (only scaled) under the linear transformation represented by the matrix. This makes them incredibly useful for understanding the behavior of complex systems.

How to Calculate Eigenvalues

Finding eigenvalues involves solving the characteristic equation:

Characteristic Equation

$$\det(A - \lambda I) = 0$$

The step-by-step process:

Form the matrix (A - λI): Subtract λ times the identity matrix from A
Calculate the determinant: Find det(A - λI), which gives the characteristic polynomial
Solve the polynomial: Set the determinant equal to zero and solve for λ
The solutions are eigenvalues: Each root of the characteristic polynomial is an eigenvalue

Example: 2×2 Matrix

For a 2×2 matrix, the characteristic polynomial is always quadratic:

2×2 Characteristic Polynomial

$$\lambda^2 - \text{trace}(A)\lambda + \det(A) = 0$$

How to Calculate Eigenvectors

For each eigenvalue λ, find the corresponding eigenvector by solving:

Eigenvector Equation

$$(A - \lambda I)\mathbf{v} = \mathbf{0}$$

This is a homogeneous system of linear equations. The eigenvector v is any non-zero vector in the null space (kernel) of (A - λI). Note that eigenvectors are not unique; any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.

How to Use This Calculator

Select matrix size: Choose 2×2 or 3×3 matrix
Enter matrix elements: Input values (integers, decimals, or fractions like 1/2)
Click Calculate: The calculator will compute eigenvalues and eigenvectors
Review results: Examine eigenvalues, eigenvectors, matrix properties, and visualization
Study the steps: Follow the detailed step-by-step solution to understand the process

Applications of Eigenvalues and Eigenvectors

Principal Component Analysis (PCA)

In data science, eigenvectors of the covariance matrix define principal components for dimensionality reduction.

Quantum Mechanics

Observable quantities correspond to eigenvalues of Hermitian operators; eigenvectors represent quantum states.

Vibration Analysis

Natural frequencies of mechanical systems are eigenvalues; mode shapes are eigenvectors.

Google PageRank

The PageRank algorithm uses the dominant eigenvector of the web link matrix to rank pages.

Differential Equations

Systems of linear ODEs are solved using eigenvalues and eigenvectors of the coefficient matrix.

Image Compression

Eigenfaces and singular value decomposition use eigenvectors for efficient image representation.

Key Properties of Eigenvalues

Sum of eigenvalues equals trace: λ₁ + λ₂ + ... + λₙ = trace(A)
Product of eigenvalues equals determinant: λ₁ × λ₂ × ... × λₙ = det(A)
Symmetric matrices have real eigenvalues: All eigenvalues of a symmetric matrix are real numbers
Complex eigenvalues come in conjugate pairs: For real matrices, complex eigenvalues occur as a ± bi
Zero eigenvalue indicates singularity: A matrix is singular (non-invertible) if and only if it has zero as an eigenvalue

Matrix Definiteness

For symmetric matrices, eigenvalues determine definiteness:

Positive definite: All eigenvalues > 0
Positive semi-definite: All eigenvalues ≥ 0
Negative definite: All eigenvalues < 0
Negative semi-definite: All eigenvalues ≤ 0
Indefinite: Mix of positive and negative eigenvalues

Frequently Asked Questions

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. For a square matrix A, an eigenvector v is a non-zero vector that, when multiplied by A, results in a scalar multiple of itself: Av = λv. The scalar λ is called the eigenvalue. Geometrically, eigenvectors point in directions that remain unchanged (only scaled) under the linear transformation represented by the matrix.

How do you find eigenvalues?

To find eigenvalues: 1) Form the matrix (A - λI) where I is the identity matrix. 2) Set the determinant det(A - λI) = 0, which gives the characteristic polynomial. 3) Solve this polynomial equation for λ. The solutions are the eigenvalues of matrix A.

How do you find eigenvectors?

For each eigenvalue λ, find the eigenvector by solving the homogeneous system (A - λI)v = 0. This means finding vectors in the null space of (A - λI). The solution gives the eigenvector direction; any non-zero scalar multiple is also an eigenvector for the same eigenvalue.

What is the characteristic polynomial?

The characteristic polynomial of a matrix A is det(A - λI), where λ is a variable and I is the identity matrix. For a 2×2 matrix, this gives a quadratic polynomial; for a 3×3 matrix, a cubic polynomial. The roots of this polynomial are the eigenvalues of A.

What are eigenvalues used for?

Eigenvalues and eigenvectors have numerous applications: solving systems of differential equations, Principal Component Analysis (PCA) in data science, Google's PageRank algorithm, quantum mechanics (observables and states), vibration analysis in engineering, stability analysis of dynamical systems, and image compression.

Can eigenvalues be complex numbers?

Yes, eigenvalues can be complex numbers, especially for non-symmetric matrices. However, symmetric matrices always have real eigenvalues. Complex eigenvalues always occur in conjugate pairs for matrices with real entries. Complex eigenvalues often indicate rotational components in the transformation.

What are the eigenvalues of a 2×2 diagonal matrix with entries 3 and 5?

For the matrix with rows [3, 0] and [0, 5], the eigenvalues are 3 and 5. Each coordinate axis is an eigenvector direction: multiplying [1, 0] by the matrix scales it by 3, while multiplying [0, 1] scales it by 5. A diagonal matrix makes this example especially direct because its diagonal entries are its eigenvalues.

What does a zero eigenvalue tell me about a matrix?

A square matrix has a zero eigenvalue exactly when it is singular, meaning it has no inverse. Geometrically, the matrix collapses at least one nonzero direction to the zero vector. For example, a matrix that maps every vector onto a line loses information about directions perpendicular to that line and therefore has determinant zero.

Can I enter fractions such as 1/2 in a matrix?

Yes. The input instructions allow fractions such as 1/2, as well as decimal values. Enter each row on its own line in the pasted matrix field, separating the entries with spaces or commas. For example, the rows “1/2, 0” and “0, 3” represent a diagonal matrix with entries one-half and 3. Use a square matrix because eigenvalues are defined for square matrices.

Additional Resources

Reference this content, page, or tool as:

"Eigenvalue and Eigenvector Calculator" at https://MiniWebtool.com/eigenvalue-eigenvector-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 22, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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