# Matrix Eigenvalues Calculator

> Find all eigenvalues of a square matrix up to 10 × 10, real and complex, with the exact characteristic polynomial and exact multiplicities.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/matrix-eigenvalues-calculator

Type the matrix with one row per line. The characteristic polynomial det(λI − A) is computed exactly, and its roots are the eigenvalues, complex ones included.

Also called: Characteristic polynomial calculator, Eigenvalue finder.

## Inputs
- Matrix A (default 4 1 2
0 3 1
1 0 2)

## Results
- Eigenvalues λ
- Characteristic polynomial det(λI − A)
- Trace (sum of eigenvalues)
- Determinant (product of eigenvalues)
- Spectral radius max |λ|
- Real eigenvalues (with multiplicity)

## Formula

```
p(λ) = det(λI − A) by the Faddeev–LeVerrier recursion: Mₖ = A·Mₖ₋₁ + cₙ₋ₖ₊₁·I,  cₙ₋ₖ = −tr(A·Mₖ) ÷ k;  eigenvalues are the roots of p(λ)
```

Where:
- A: the n × n matrix
- I: the identity matrix
- cₖ: coefficients of the characteristic polynomial
- λ: an eigenvalue

## Worked example
1. Trace = 4 + 3 + 2 = 9 and det A = 4(3·2 − 1·0) − 1(0·2 − 1·1) + 2(0·0 − 3·1) = 24 + 1 − 6 = 19.
2. The sum of the 2 × 2 principal minors is (12 − 0) + (8 − 2) + (6 − 0) = 24, so p(λ) = λ³ − 9λ² + 24λ − 19.
3. Its roots are λ ≈ 1.4679, 2.6527 and 4.8794; they add to 9 and multiply to 19.

**Result:** The matrix has three real eigenvalues, about 1.468, 2.653 and 4.879.

## What the result means

Eigenvalues tell you how the matrix stretches space along its eigenvectors. Complex eigenvalues of a real matrix come in conjugate pairs and indicate rotation. The spectral radius decides whether repeated multiplication by A grows (> 1) or dies out (< 1).

## Assumptions and limits
- Entries are real numbers, read exactly as decimals or fractions.
- Up to 10 × 10.
- Eigenvectors are not computed.
- Eigenvalues are numerical (10 significant figures); the characteristic polynomial is exact.

## Questions

### What does multiplicity mean here?

It is the algebraic multiplicity: how many times λ is a root of the characteristic polynomial. It is computed exactly, so a repeated eigenvalue is not reported as two nearly equal numbers.

### Why are the eigenvalues of a symmetric matrix real?

The spectral theorem: a real symmetric matrix can be diagonalised by an orthogonal matrix, so all its eigenvalues are real.

## Related calculators
- [Matrix Calculator](https://calculators.geekmidst.com/calculators/matrix-calculator)
- [Polynomial Roots Calculator](https://calculators.geekmidst.com/calculators/polynomial-roots-calculator)
- [Gaussian Elimination Calculator](https://calculators.geekmidst.com/calculators/linear-system-gaussian-elimination-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/matrix-eigenvalues-calculator