Eigenvalue Matrix Calculator

Calculator Input

Matrix size

Decimal precision

Example loader

a11

a12

a13

a21

a22

a23

a31

a32

a33

Example Data Table

Use this sample to test the calculator quickly.

Matrix	Values	Expected eigenvalues	Note
2 x 2 diagonal	[[5, 0], [0, 2]]	5, 2	Diagonal entries are eigenvalues.
2 x 2 rotation	[[0, -1], [1, 0]]	i, -i	Complex roots show rotation.
3 x 3 triangular	[[4, 2, 1], [0, 3, -1], [0, 0, 2]]	4, 3, 2	Triangular diagonal gives eigenvalues.

Formula Used

Eigenvalue definition: A v = λ v, where v ≠ 0.

Characteristic equation: det(λI - A) = 0.

2 x 2 matrix: λ² - trace(A)λ + det(A) = 0.

3 x 3 matrix: λ³ - trace(A)λ² + c₂λ - det(A) = 0.

Second invariant: c₂ = a₁₁a₂₂ + a₁₁a₃₃ + a₂₂a₃₃ - a₁₂a₂₁ - a₁₃a₃₁ - a₂₃a₃₂.

The calculator solves the 2 x 2 equation directly. For 3 x 3 matrices, it solves the cubic equation with a complex iterative method.

How to Use This Calculator

Select a 2 x 2 or 3 x 3 matrix.
Enter each matrix value in the labeled cell fields.
Choose the decimal precision for the final output.
Press the calculate button to show eigenvalues and vectors.
Review the graph, formula, trace, determinant, and polynomial.
Download the CSV or PDF file when you need a record.

Understanding Eigenvalues

Eigenvalues describe how a square matrix stretches special directions. Those directions are called eigenvectors. When a matrix multiplies an eigenvector, the direction stays the same. Only the length and sign may change. This calculator focuses on 2 by 2 and 3 by 3 matrices. These sizes cover many classroom, engineering, data, and graphics tasks.

Why Matrix Eigenvalues Matter

Eigenvalues help explain stability, vibration, growth, rotation, and repeated transformations. A positive eigenvalue keeps direction. A negative eigenvalue reverses direction. A complex eigenvalue often signals rotation with scaling. In systems of differential equations, eigenvalues show whether a solution grows, decays, or oscillates. In data analysis, they support principal component methods. In physics, they help solve energy and mode problems.

How This Calculator Works

The tool builds the characteristic polynomial from your matrix. For a 2 by 2 matrix, it uses the trace, determinant, and quadratic formula. For a 3 by 3 matrix, it builds the cubic characteristic equation. It then applies an iterative complex root method. This lets the calculator report real and complex eigenvalues in one place. It also estimates eigenvectors when the eigenvalue is real.

Reading The Results

The real part shows horizontal position on the graph. The imaginary part shows vertical position. Points on the horizontal axis are real eigenvalues. Points above or below it are complex values. The determinant equals the product of eigenvalues. The trace equals their sum. These checks help you verify the result quickly. Small rounding errors can appear because some roots are numeric.

Practical Tips

Enter clean numbers first. Avoid very large mixed scales when possible. Use higher precision when values are close together. Check the characteristic polynomial before using the result in a report. Download the CSV file for spreadsheet work. Use the PDF file for sharing, printing, or lesson notes. If the matrix has repeated eigenvalues, eigenvectors may need extra manual checking.

Use Cases

Teachers can demonstrate roots, traces, and determinants together. Students can compare hand solutions with numeric output. Engineers can study modes and stability. Developers can test transformation matrices before coding simulations or visual tools. This workflow saves time while keeping algebra visible, structured, and easier to review.

FAQs

What is an eigenvalue?

An eigenvalue is a number that shows how a matrix scales an eigenvector. The eigenvector keeps its direction after multiplication by the matrix.

What matrix sizes are supported?

This calculator supports 2 x 2 and 3 x 3 square matrices. These formats cover many common algebra, physics, engineering, and data tasks.

Can eigenvalues be complex?

Yes. Matrices can have complex eigenvalues. They often appear when the matrix includes rotation effects, oscillation behavior, or a negative discriminant.

Why does the calculator show a graph?

The graph places each eigenvalue by real and imaginary parts. It helps you see whether roots are real, complex, repeated, or balanced around an axis.

Are eigenvectors always displayed?

Real eigenvectors are estimated for real eigenvalues. Complex eigenvectors are not displayed here, because they need complex vector formatting and extra interpretation.

What does trace mean?

The trace is the sum of diagonal matrix entries. For a square matrix, it also equals the sum of eigenvalues when multiplicities are counted.

What does determinant mean?

The determinant measures scale and orientation effects. It also equals the product of all eigenvalues, counted with their algebraic multiplicities.

Why are some answers rounded?

Some 3 x 3 roots need numerical solving. The precision setting controls displayed decimals, but tiny rounding differences can still appear.