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Walk-through

How to Use This Calculator

5 steps

1

Choose a matrix size

Pick 2×2 or 3×3 from the size selector on the Eigenvalues tab. Switching sizes swaps in a fresh entry grid — your 2×2 entries and your 3×3 entries are kept separately, so you can flip back and forth without losing either.

2

Enter the matrix

Type each entry of the matrix, row by row, into the grid. The preview line above the grid ("A = [...]") echoes the matrix back so you can double-check you typed it correctly before reading the result.

3

Read the eigenvalues

The result card on the Eigenvalues tab lists every λ that solves det(A − λI) = 0, along with whether each is real or complex and how many times it repeats (its multiplicity). Real matrices can still have complex eigenvalues — the calculator handles both.

4

Check the eigenvectors

Switch to the Eigenvectors tab to see the unit eigenvector paired with each eigenvalue, solved from the null space of (A − λI). If an eigenvalue repeats and its eigenspace turns out to be more than one dimension wide, the interpretation line below the table calls that out explicitly.

5

Try the presets

Use the preset chips to jump straight to a worked example of each case for both matrix sizes — a real, diagonalizable matrix; a matrix with complex conjugate eigenvalues; and a matrix with a repeated eigenvalue — so you can see how the solver handles each situation before you enter your own matrix.

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Reference

Formula & Methodology

4 formulas

Characteristic equation

det(A − λI) = 0

Every eigenvalue λ of a square matrix A is a root of this equation, where I is the identity matrix of the same size. Subtracting λ from each diagonal entry of A and setting the determinant to zero turns the eigenvalue problem into a polynomial root-finding problem.

2×2 characteristic polynomial

λ² − tr(A)·λ + det(A) = 0, where tr(A) = a + d, det(A) = ad − bc for A = [[a, b], [c, d]]

For a 2×2 matrix the characteristic equation reduces to a quadratic in λ, built entirely from the matrix's trace (the sum of its diagonal entries) and its determinant. The calculator solves this directly with the quadratic formula, branching to a complex-conjugate pair when the discriminant is negative.

3×3 characteristic polynomial

λ³ − tr(A)·λ² + M₂(A)·λ − det(A) = 0, where M₂(A) is the sum of the three principal 2×2 minors of A

For a 3×3 matrix the characteristic equation is a cubic in λ. The calculator reuses the Cubic Equation Calculator's tested cubic solver to find its roots — one real root and a complex conjugate pair, or three real roots, depending on the matrix.

Eigenvector (null space of A − λI)

(A − λI)v = 0, v ≠ 0

Once an eigenvalue λ is known, its eigenvectors are exactly the nonzero vectors v that (A − λI) sends to zero — the null space of that matrix. The calculator finds one such vector per eigenvalue by Gaussian elimination, then scales it to unit length.

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Glossary

Key Terms Explained

7 terms

Eigenvalue A scalar λ for which there exists a nonzero vector v satisfying Av = λv — multiplying the matrix by v just scales v, without changing its direction. An n×n matrix has exactly n eigenvalues counted with multiplicity, though some may be complex or repeated.

Eigenvector A nonzero vector v that a matrix A only stretches or shrinks (never rotates), satisfying Av = λv for the paired eigenvalue λ. Eigenvectors are only defined up to scaling — this calculator reports the unit-length version of each one.

Characteristic polynomial The polynomial det(A − λI), obtained by subtracting λ from each diagonal entry of A and expanding the determinant. Its roots are exactly the matrix's eigenvalues — a quadratic for a 2×2 matrix, a cubic for a 3×3 matrix.

Trace The sum of a matrix's diagonal entries, tr(A) = a₁₁ + a₂₂ + ... + aₙₙ. The trace always equals the sum of the eigenvalues, counted with multiplicity — a quick way to sanity-check a hand calculation.

Determinant A single number computed from a square matrix that, among other things, always equals the product of its eigenvalues (counted with multiplicity). A zero determinant means at least one eigenvalue is zero, and the matrix has no inverse.

Diagonalizable A matrix is diagonalizable when it has enough independent eigenvectors to form a full basis — equivalently, when every repeated eigenvalue's eigenspace dimension (geometric multiplicity) matches how many times it repeats (algebraic multiplicity). A matrix that fails this test is called defective.

Null space The set of all vectors that a matrix sends to the zero vector. Eigenvectors for a given eigenvalue λ are found as nonzero vectors in the null space of (A − λI) — everything that matrix maps to zero.

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Scenarios

Real-World Examples

3 worked examples

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Basic 2×2 case

A = [[4, 1], [2, 3]]

Matrix size 2×2 Matrix [[4, 1], [2, 3]]

The trace is 7 and the determinant is 10, so the characteristic equation is λ² − 7λ + 10 = 0, which factors cleanly to give λ = 5 and λ = 2 — two distinct real eigenvalues, matching the fact that a real symmetric-ish 2×2 matrix like this one almost always diagonalizes over the reals.

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Reading the eigenvectors

Same matrix, Eigenvectors tab

Matrix size 2×2 Matrix [[4, 1], [2, 3]]

For λ = 5, solving (A − 5I)v = 0 gives the unit eigenvector (0.7071, 0.7071) — pointing along the direction (1, 1). For λ = 2, the eigenvector is (−0.4472, 0.8944), pointing along (−1, 2). Multiplying A by either vector reproduces the same vector scaled by its eigenvalue, which is exactly what Av = λv means.

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Complex eigenvalues

A = [[0, −1], [1, 0]] (a 90° rotation matrix)

Matrix size 2×2 Matrix [[0, -1], [1, 0]]

This matrix rotates every real vector by 90°, so no real vector keeps its direction — its eigenvalues are the complex conjugate pair λ = i and λ = −i, with the characteristic equation simplifying to λ² + 1 = 0. This is a common pattern: rotation matrices (and other matrices with no real fixed direction) always produce complex eigenvalues.

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Deep Dive

Understanding the Eigenvalue Calculator

3 chapters · 5 min read

Eigenvalues and eigenvectors describe the directions a matrix leaves unchanged (aside from scaling) — a concept that shows up everywhere from vibration analysis and quantum mechanics to Google's PageRank algorithm and principal component analysis in statistics. This calculator solves for both directly from a 2×2 or 3×3 matrix you enter, showing the characteristic polynomial, the eigenvalues (real or complex), and a unit eigenvector for each.

How the solver works

The calculator first builds the characteristic polynomial det(A − λI) from the matrix's trace and determinant (and, for a 3×3 matrix, the sum of its principal 2×2 minors). For a 2×2 matrix this polynomial is a quadratic, solved directly with the quadratic formula. For a 3×3 matrix it's a cubic, solved with the same tested cubic solver used by the Cubic Equation Calculator — the characteristic polynomial of any 3×3 matrix is, after all, just a cubic equation in λ. Once the eigenvalues are known, the calculator finds an eigenvector for each one by row-reducing (A − λI) to find a nonzero vector in its null space, then scales that vector to unit length.

Why eigenvalues can be complex — and what repeated eigenvalues mean

Even a matrix built entirely from real numbers can have complex eigenvalues, whenever it rotates space rather than just stretching it along some real direction (the 90° rotation matrix in the examples above is the cleanest case). When that happens, the complex eigenvalues always arrive as a conjugate pair, just like the complex roots of a real polynomial. Separately, a matrix can have a repeated eigenvalue — the same λ satisfying the characteristic equation more than once. If that eigenvalue's eigenspace (the set of all its eigenvectors) is exactly as many dimensions wide as the repeat count, the matrix is still diagonalizable. If the eigenspace is narrower — only one independent direction even though the eigenvalue repeats twice or three times — the matrix is called defective, and the calculator's interpretation line flags this case specifically.

Where this shows up in practice

Eigenvalues measure how much a matrix stretches space along its eigenvector directions, which is why they appear across such a wide range of applications: a structure's natural vibration frequencies are the eigenvalues of its stiffness matrix, a Markov chain's long-run behavior is governed by the eigenvector tied to its largest eigenvalue, and principal component analysis in statistics ranks directions in a dataset by the eigenvalues of its covariance matrix. In every case, the same underlying question is being asked: which directions does this matrix treat as "special," and by how much does it scale each one?

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Questions

Frequently Asked Questions

6 questions

What formula does the Eigenvalue Calculator use?+

It solves the characteristic equation det(A − λI) = 0 — a quadratic in λ for a 2×2 matrix, a cubic in λ for a 3×3 matrix — using the matrix's trace, determinant, and (for 3×3) the sum of its principal 2×2 minors.

How are the eigenvectors computed?+

For each eigenvalue λ, the calculator row-reduces (A − λI) with Gaussian elimination to find a nonzero vector in its null space — the vectors that (A − λI) sends to zero — then scales that vector to unit length.

Can a real matrix have complex eigenvalues?+

Yes. Any matrix built entirely from real numbers can still have complex eigenvalues — this happens whenever the matrix rotates space rather than stretching it along a real direction. Complex eigenvalues of a real matrix always occur in conjugate pairs, such as λ = a + bi and λ = a − bi.

What does the trace tell me?+

The trace — the sum of a matrix's diagonal entries — always equals the sum of its eigenvalues, counted with multiplicity. It's a fast way to sanity-check an eigenvalue calculation by hand: if your eigenvalues don't add up to the trace, something went wrong.

What does a repeated eigenvalue mean?+

It means the same λ satisfies the characteristic equation more than once (its algebraic multiplicity is greater than one). Whether the matrix is still diagonalizable depends on whether that eigenvalue's eigenspace is equally wide — if it is, there are that many independent eigenvector directions; if it's narrower, the matrix is called defective and only has one independent eigenvector for that eigenvalue.

Does this calculator support matrices larger than 3×3?+

No — it's built for 2×2 and 3×3 matrices, where the characteristic polynomial is a quadratic or cubic that can be solved with a closed-form formula. Matrices 4×4 and larger generally require numerical eigenvalue algorithms instead, since there's no general closed-form solution for polynomials of degree 5 or higher.

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Eigenvalue Calculator

Matrix

Eigenvalues

Eigenvectors

Eigenvalue → eigenvector

Characteristic polynomial

This matrix's characteristic polynomial

How to Use This Calculator

Choose a matrix size

Enter the matrix

Read the eigenvalues

Check the eigenvectors

Try the presets

Formula & Methodology

Key Terms Explained

Real-World Examples

Basic 2×2 case

Reading the eigenvectors

Complex eigenvalues

Understanding the Eigenvalue Calculator

How the solver works

Why eigenvalues can be complex — and what repeated eigenvalues mean

Where this shows up in practice

Frequently Asked Questions

Get a branded PDF of your results

Eigenvalue Calculator

Matrix

Eigenvalues

Eigenvectors

Eigenvalue → eigenvector

Characteristic polynomial

This matrix's characteristic polynomial

How to Use This Calculator

Choose a matrix size

Enter the matrix

Read the eigenvalues

Check the eigenvectors

Try the presets

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Eigenvalue Calculator

How the solver works

Why eigenvalues can be complex — and what repeated eigenvalues mean

Where this shows up in practice

Frequently Asked Questions

Get a branded PDF of your results

Keep Exploring

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Guides & articles