How do I calculate eigenvalues by hand?

To find eigenvalues: 1) Form the characteristic equation det(A - λI) = 0. 2) Expand the determinant to get a polynomial in λ. 3) Solve the polynomial. For a 2×2 matrix, this gives a quadratic equation; for 3×3, a cubic equation.

Can eigenvalues be complex numbers?

Yes! When the discriminant of the characteristic polynomial is negative, eigenvalues come in complex conjugate pairs (a + bi and a - bi). This is common for rotation matrices and oscillatory systems.

What is the relationship between eigenvalues and matrix properties?

Eigenvalues encode key matrix properties: The sum of eigenvalues equals the trace. The product of eigenvalues equals the determinant. A matrix is invertible if and only if all eigenvalues are non-zero.

What matrix sizes does this calculator support?

This eigenvalue calculator supports square matrices from 2×2 up to 6×6. For 2×2 and 3×3 matrices, we provide exact analytical solutions. For larger matrices, we use the numerical QR algorithm.

Linear Algebra

Online Eigenvalue Calculator

Compute eigenvalues of square matrices with step-by-step explanations.

100% Free2×2, 3×3, largerWith Steps

Matrix Configuration

Set the matrix size (square matrices only for eigenvalues)

Size:

3×3

Matrix Input

Enter matrix elements (decimals and fractions allowed)

The Characteristic Polynomial Approach

How eigenvalues are derived analytically from the characteristic equation

For a 2×2 matrix

Setting $\det(A - \lambda I) = 0$ expands to a quadratic:

\det(A - \lambda I) = (a-\lambda)(d-\lambda) - bc = 0

\lambda^2 - (a+d)\lambda + (ad - bc) = 0

Solved with the quadratic formula. The discriminant $\Delta = (a+d)^2 - 4(ad-bc)$ determines whether eigenvalues are real ( $\Delta \geq 0$ ) or complex ( $\Delta < 0$ ).

For a 3×3 matrix

The characteristic polynomial is a degree-3 (cubic) equation:

-\lambda^3 + \text{tr}(A)\,\lambda^2 - \frac{1}{2}\left[(\text{tr}\,A)^2 - \text{tr}(A^2)\right]\lambda + \det(A) = 0

This calculator solves it using Cardano's method (depressed cubic). Three cases arise: one real root (positive discriminant), a repeated real root (zero discriminant), or three distinct real roots (negative discriminant — the casus irreducibilis requiring trigonometric solution).

For n > 3 matrices

No closed-form formula exists (Abel–Ruffini theorem). This calculator uses the QR algorithm: iteratively decomposing $A = QR$ and updating $A \leftarrow RQ$ until the matrix converges to (quasi-)triangular form, reading eigenvalues from the diagonal.

Real-World Applications

Where eigenvalues appear across science and engineering

Principal Component Analysis (PCA)

Eigenvalues of the covariance matrix rank the importance of each principal component. The largest eigenvalue corresponds to the direction of greatest variance in the data.

Google PageRank

PageRank is the dominant eigenvector of the web's link-transition matrix, corresponding to eigenvalue 1. The power method iteratively finds this eigenvector.

Quantum Mechanics

Observable quantities (energy, momentum, spin) are eigenvalues of Hermitian operators. Schrödinger's equation $H\psi = E\psi$ is an eigenvalue problem where E is the energy eigenvalue.

Structural Engineering

Natural frequencies of vibration of a structure are square roots of eigenvalues of $M^{-1}K$ (mass-stiffness system). Engineers must ensure these don't match external forcing frequencies to avoid resonance.

Matrix Types and Their Eigenvalue Properties

Quick reference for common matrix classes

Matrix Type	Definition	Eigenvalue Property
Symmetric	$A = A^T$	All eigenvalues are real
Orthogonal	$A^T A = I$	$\|\lambda\| = 1$ (on unit circle)
Positive Definite	$x^T A x > 0$	All eigenvalues $\lambda > 0$
Skew-Symmetric	$A = -A^T$	Purely imaginary or zero
Diagonal	Off-diagonal entries = 0	Eigenvalues are the diagonal entries
Idempotent	$A^2 = A$	Each eigenvalue is 0 or 1
Nilpotent	$A^k = 0$	All eigenvalues are 0

Example Matrices

Click on any example to load it into the calculator

2×2 Basic Example

Simple 2×2 matrix with real eigenvalues

[2, 1]

[1, 3]

3×3 Example

3×3 matrix with mixed eigenvalues

[1, 2, 3]

[3, 2, 1]

[2, 1, 3]

Identity Matrix

3×3 identity matrix (eigenvalues = 1)

[1, 0, 0]

[0, 1, 0]

[0, 0, 1]

Symmetric Matrix

Symmetric matrix with real eigenvalues

[4, 1, 2]

[1, 3, 0]

[2, 0, 5]

Learn More About Eigenvalues

MIT OpenCourseWare

Linear algebra lectures on eigenvalues and eigenvectors by Prof. Strang

3Blue1Brown

Essence of Linear Algebra video series with visual intuition

Wolfram MathWorld

Detailed mathematical reference on eigenvalue theory and computation

Frequently Asked Questions

Eigenvalues are special scalars λ that satisfy the equation Av = λv for a matrix A and non-zero vector v (called an eigenvector). They reveal intrinsic properties of linear transformations, showing how the matrix scales vectors along certain directions. Eigenvalues are fundamental in physics (vibrational modes, quantum mechanics), engineering (stability analysis), and data science (PCA, machine learning).

RREF Calculator

Linear Algebra

Reduce matrices to row echelon form step-by-step

Matrix Multiplication Calculator

Linear Algebra

Multiply matrices with detailed steps

Cross Product Calculator

Vectors

Calculate cross product of 3D vectors

Quadratic Formula Calculator

Algebra

Solve quadratic equations step-by-step