Eigenvector Calculator

Matrix Input Form

Matrix size

Decimal precision

Vector normalization

Matrix Entries

For a 2 by 2 matrix, only the first four entries are used.

a11

a12

a13

a21

a22

a23

a31

a32

a33

Example Data Table

Example	Matrix	Expected eigenvalues	Common basis idea
Diagonal 2 by 2	[[2, 0], [0, 3]]	2, 3	Standard coordinate vectors
Symmetric 2 by 2	[[4, 1], [1, 4]]	5, 3	Opposite and matching directions
Triangular 3 by 3	[[1, 2, 0], [0, 3, 1], [0, 0, 4]]	1, 3, 4	Row reduction gives each space

Formula Used

The calculator uses the eigenvector equation A v = λ v. This can be rearranged as (A - λI)v = 0.

For a 2 by 2 matrix, the characteristic equation is λ² - trace(A)λ + det(A) = 0.

For a 3 by 3 matrix, the characteristic equation is λ³ - c1λ² + c2λ - c3 = 0.

Here, c1 is trace(A), c2 is one half of trace(A) squared minus trace(A²), and c3 is det(A).

After each eigenvalue is found, row reduction solves the null space of A - λI. Any nonzero null-space vector is an eigenvector.

How To Use This Calculator

Select a 2 by 2 or 3 by 3 matrix. Enter every visible coefficient in the matrix entry boxes.

Choose a decimal precision and vector normalization style. Unit length is useful for geometry and comparison.

Press the calculate button. The result section will appear above the form and below the page header.

Review the trace, determinant, characteristic equation, eigenvalues, and eigenvector basis. Use the export buttons to save your results.

Why This Eigenvector Tool Helps

Eigenvectors show stable directions inside a linear transformation. When a matrix acts on an eigenvector, the direction remains the same. Only the scale changes. That scale is the eigenvalue. This calculator gives a structured way to study that idea. It accepts two by two and three by three matrices. It then builds the characteristic equation, finds eigenvalues, and solves the matching vector spaces.

Practical Matrix Work

Many learners can compute a determinant, yet still lose track during row reduction. This page keeps the work organized. The result panel appears immediately after submission. It lists trace, determinant, characteristic coefficients, eigenvalues, and vector bases. It also supports normalization, so vectors become easier to compare. Unit vectors help with geometry. First nonzero normalization helps with classroom answers. Maximum component normalization helps with quick checks.

Use Cases

Eigenvectors are used in engineering, physics, data analysis, graphics, economics, and systems modeling. They describe vibration modes, principal components, repeated processes, Markov behavior, and diagonalization steps. A good calculator should therefore show more than one number. It should reveal how each result was produced. This tool does that with formula notes, example data, and export buttons.

Reading The Results

A zero determinant may show that zero is an eigenvalue. Repeated eigenvalues may have one or more independent eigenvectors. A three by three matrix can also produce complex roots. This calculator focuses vector solving on real roots, because most classroom and applied examples expect real vector bases. When complex roots appear, the note explains the limitation clearly. You can still review the real root and its vector.

Study Workflow

Start with a simple diagonal matrix. Then try a triangular matrix. After that, enter a symmetric matrix. Symmetric real matrices usually give real eigenvalues and clean eigenvectors. Compare your manual work against the result table. Export the table when you need a record. The CSV file works well for spreadsheets. The PDF file is useful for assignments and teaching notes. Always check rounding precision before copying results.

Common Accuracy Notes

Rounding can change the visible vector slightly. Scaling can also change appearance without changing direction. If two answers are scalar multiples, they represent the same eigenvector direction in the eigenspace for that eigenvalue group.

FAQs

What is an eigenvector?

An eigenvector is a nonzero vector that keeps its direction when a matrix transforms it. The vector may stretch, shrink, or reverse. The scaling amount is called the eigenvalue.

What matrix sizes are supported?

This page supports 2 by 2 and 3 by 3 square matrices. These sizes cover many classroom, engineering, and data analysis examples.

Can an eigenvalue be zero?

Yes. A zero eigenvalue often appears when the determinant is zero. It means the matrix sends at least one nonzero direction toward a lower-dimensional result.

Why can eigenvectors look different?

Any nonzero scalar multiple of an eigenvector is still an eigenvector. Normalization changes the display, but not the actual eigenspace direction.

What does unit normalization mean?

Unit normalization divides the vector by its length. The final vector has length one, which makes directions easier to compare.

Does the calculator show complex results?

It shows complex eigenvalues when they appear. Two by two complex vector bases are expanded. Three by three complex vector solving is noted but not expanded.

What is the characteristic equation?

The characteristic equation comes from det(A - λI) = 0. Its roots are the eigenvalues used to find eigenvectors.

How should I check my answer?

Multiply A by the reported vector. Then multiply the vector by its eigenvalue. Matching results confirm the eigenvector relationship.