Eigenvalue Calculator

Solve det(A − λI) = 0 → λ² − (tr A)λ + det A = 0 step by step — free eigenvalue calculator with worked examples, real-world defaults and instant answers.

det(A − λI) = 0 → λ² − (tr A)λ + det A = 0

Matrix A (2×2)

λ₁2

λ₂5

v₁ (direction)[1, -2]

v₂ (direction)[1, 1]

1
Trace and determinant build the characteristic polynomial:
tr A = 4 + 3 = 7, det A = 10 λ² − 7λ + 10 = 0
2
Solve the quadratic:
Δ = 7² − 4(10) = 9 → λ = 2, λ = 5
3
For each λ, solve (A − λI)v = 0 for an eigenvector direction:
λ = 2: v = [1, -2] λ = 5: v = [1, 1]
4
Checks: eigenvalues sum to the trace and multiply to the determinant:
2 + 5 = 7 ✓, product = 10 ✓

Eigenvectors are the directions a matrix merely stretches (by the eigenvalue) without rotating — the skeleton of the transformation. They drive PCA, vibration modes, Google's PageRank and stability analysis. The trace/determinant identities are the fastest sanity check.

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Eigenvalues & eigenvectors of a 2×2 — characteristic polynomial route.

About Eigenvalue Calculator

The Eigenvalue Calculator solves det(A − λI) = 0 → λ² − (tr A)λ + det A = 0 for you with the full working shown — every substitution, every unit, every step, exactly the way a good teacher writes it on the board. Eigenvectors are the directions a matrix merely stretches (by the eigenvalue) without rotating — the skeleton of the transformation.

How to use Eigenvalue Calculator

1Enter your known values in the input fields (sensible real-world defaults are pre-filled).
2The tool substitutes them into det(A − λI) = 0 → λ² − (tr A)λ + det A = 0 and recomputes live.
3Read the answer in the result box, then expand the step-by-step solution to see the full working.
4Copy the method into your notebook, or change inputs to explore how the result behaves.

Why use Eigenvalue Calculator?

✓Solves det(A − λI) = 0 → λ² − (tr A)λ + det A = 0 instantly as you type — no submit button, no waiting
✓Step-by-step solution shown for every calculation, not just the final answer
✓Realistic example values pre-loaded so you can see a worked example immediately
✓100% free, no sign-up, runs entirely in your browser — your numbers never leave your device
✓Mobile-friendly and fast enough to use mid-homework or mid-lesson

Frequently asked questions

What formula does the Eigenvalue Calculator use?+

It uses det(A − λI) = 0 → λ² − (tr A)λ + det A = 0. The steps section shows the formula with your actual numbers substituted, so you can follow (and verify) every stage of the calculation rather than trusting a black box.

What should I keep in mind when using this calculator?+

Is the Eigenvalue Calculator free to use?+

Completely free, with no sign-up or limits. All computation happens client-side in your browser, so it is fast, private, and works even on slow connections once the page has loaded.

Can I use this for homework, teaching or exam prep?+

That is exactly what it is built for. The step-by-step working mirrors how solutions are presented in class, so you can check homework, build worked examples for teaching, or practise method recall before exams.

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