Matrix Eigenvalues Calculator

Find eigenvalues via the characteristic polynomial (2×2, 3×3) or QR iteration (larger symmetric).

About Matrix Eigenvalues Calculator

Eigenvalues λ satisfy det(A − λI) = 0. For 2×2 matrices the characteristic polynomial is quadratic (closed form). For 3×3 it is a cubic solved via Cardano's formula. For larger matrices, QR iteration is used — this implementation supports symmetric matrices of any size (which always have real eigenvalues). Non-symmetric matrices larger than 3×3 are not supported. Complex eigenvalues are reported when the discriminant is negative.

Formulas

Characteristic Equation

det(A − λI) = 0

A: Square matrix
λ: Eigenvalue
I: Identity matrix

2×2 Formula

λ = (tr(A) ± √(tr²−4·det)) / 2

tr: Trace a+d
det: Determinant ad−bc

Worked example

With the default values below, Matrix Eigenvalues Calculator returns Eigenvalues of 5 2. Change any input above and every figure here is recalculated in your browser.

Inputs used

Matrix A (rows separated by newlines or semicolons; values by commas or spaces): 4 1 2 3

Results

Eigenvalues: 5 2
Method Used: Characteristic polynomial (2×2 closed form)
Complex Eigenvalues?: No

Step by step

1Matrix A= 4 1 2 3
2Characteristic equation
3tr(A) = 7, det(A) = 10
4Eigenvalues= 5 2

Assumptions and limits

The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What matrix sizes are supported?

2×2 and 3×3 use exact closed-form solutions and report complex eigenvalues. For 4×4 and larger, only symmetric matrices are supported via QR iteration (all eigenvalues are real).

Can eigenvalues be complex?

Yes. Real matrices can have complex eigenvalues that come in conjugate pairs. This calculator reports them for 2×2 and 3×3 matrices.