Eigenvalue Calculator

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Enter a matrix and press Find eigenvalues.

This free eigenvalue calculator finds the eigenvalues and eigenvectors of any 2×2 or 3×3 matrix. Enter your matrix and it solves the characteristic equation for you — handling real and complex eigenvalues.

eigenvalue calculator solving the characteristic equation of a matrix
Eigenvalues are the scalars that stretch their eigenvectors without changing direction.

How to use the eigenvalue calculator

  1. Pick a size — 2×2 or 3×3.
  2. Type your matrix entries.
  3. Press “Find eigenvalues.” You get each eigenvalue and, for real eigenvalues, an eigenvector.

What is an eigenvalue?

For a square matrix $A$, a non-zero vector $v$ is an eigenvector and $\lambda$ its eigenvalue when multiplying by $A$ only scales the vector:

$$A\,v = \lambda\, v$$

In words: $A$ does not rotate $v$, it just stretches or shrinks it by the factor $\lambda$. Those special directions and their scale factors reveal the deepest structure of the matrix.

How eigenvalues are found

The eigenvalues are the roots of the characteristic equation, formed by subtracting $\lambda$ down the diagonal and setting the determinant to zero:

$$\det(A – \lambda I) = 0$$

For a 2×2 matrix this is a quadratic; for a 3×3 it is a cubic. The calculator builds this polynomial from your entries and solves it exactly, then finds each eigenvector by solving $(A – \lambda I)v = 0$.

💡 Two quick checks The eigenvalues always sum to the trace (the diagonal total) and multiply to the determinant. Use these to sanity-check any result.

Worked example (2×2)

For $A = \begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}$ the trace is 4 and the determinant is 3, so the characteristic equation is $\lambda^2 – 4\lambda + 3 = 0$, giving $\lambda = 3$ and $\lambda = 1$. Their eigenvectors are $[1, 1]$ and $[1, -1]$ — exactly what the calculator returns for the default matrix.

🤖 Why eigenvalues matter in machine learning

Eigenvalues are the engine of Principal Component Analysis (PCA): the eigenvectors of the covariance matrix are the principal directions, and their eigenvalues measure how much variance each captures. They also drive spectral clustering and PageRank. Build up to it with the determinant calculator and the eigenvalues of a 2×2 matrix guide.

Frequently asked questions

What does the eigenvalue calculator do?
It finds the eigenvalues and eigenvectors of a 2×2 or 3×3 matrix by building and solving the characteristic equation det(A – lambda I) = 0, including complex eigenvalues.
How do you find eigenvalues by hand?
Subtract lambda from each diagonal entry, take the determinant, set it to zero to get the characteristic polynomial, and solve for lambda. Then solve (A – lambda I)v = 0 for each eigenvector.
Can a matrix have complex eigenvalues?
Yes. Real matrices can have complex eigenvalues that occur in conjugate pairs, common for rotation matrices. The calculator reports them.
What is an eigenvector?
An eigenvector is a non-zero vector whose direction is unchanged when the matrix is applied; it is only scaled by its eigenvalue.
Do eigenvalues relate to the trace and determinant?
Yes. The eigenvalues sum to the trace and multiply to the determinant, which makes for quick sanity checks.

Repeated and complex eigenvalues

Not every matrix has neat, distinct real eigenvalues. A repeated eigenvalue occurs when the characteristic polynomial has a double (or triple) root — the matrix may then have fewer independent eigenvectors than its size, which matters for whether it can be diagonalized. Complex eigenvalues appear for matrices that rotate, such as $\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$, whose eigenvalues are $+i$ and $-i$; they always come in conjugate pairs for a real matrix. This calculator reports both cases: repeated roots are listed with their multiplicity, and complex eigenvalues are shown in $a + bi$ form. For hand calculation, remember that a 3×3 matrix always has at least one real eigenvalue, because a real cubic always has at least one real root.

Related tools

Keep exploring: Determinant Calculator, Matrix Inverse Calculator, Eigenvalues of a 2×2 Matrix, or the formal reference on Wikipedia.

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