How do you compute eigenvalues of a matrix?

To compute eigenvalues of a square matrix A, form the characteristic polynomial det(A − λI) and solve det(A − λI) = 0 for λ. The roots are the eigenvalues. For 2×2 and 3×3 matrices this can be done analytically; for larger matrices numerical methods are used.

What is the difference between algebraic and geometric multiplicity?

The algebraic multiplicity of an eigenvalue is how many times it appears as a root of the characteristic polynomial. The geometric multiplicity is the dimension of its eigenspace, i.e., the number of linearly independent eigenvectors for that eigenvalue. Geometric multiplicity is always at least 1 and at most equal to the algebraic multiplicity.

Eigenvalue & Eigenvector Calculator

Q: What are eigenvalues and eigenvectors in simple terms?

For a square matrix A, an eigenvector v is a non-zero vector whose direction does not change when A is applied: A v = λ v. The scalar λ is the corresponding eigenvalue and tells you how much the eigenvector is stretched or flipped.

Q: When is a matrix diagonalizable?

A matrix is diagonalizable if there exists a basis of eigenvectors, which happens when the sum of the geometric multiplicities of all eigenvalues equals the size of the matrix. A sufficient condition is that the matrix has n distinct eigenvalues for an n×n matrix.

Compute eigenvalues, eigenvectors, eigenspaces, and check diagonalizability for 2×2 and 3×3 matrices with full step-by-step working.

Matrix input

Matrix size: Real entries only. Complex eigenvalues will be shown in a + bi form.

Show step-by-step derivation

Results

Enter a 2×2 or 3×3 matrix and click “Compute” to see eigenvalues, eigenvectors, and eigenspaces.

How this eigenvalue & eigenvector calculator works

For a square matrix \(A\), an eigenvalue–eigenvector pair \((\lambda, \mathbf{v})\) satisfies

\[ A\mathbf{v} = \lambda \mathbf{v}, \quad \mathbf{v} \neq \mathbf{0}. \]

Rearranging gives \((A - \lambda I)\mathbf{v} = \mathbf{0}\). Non‑trivial solutions exist only when

\[ \det(A - \lambda I) = 0, \]

which is the characteristic equation. Its roots are the eigenvalues \(\lambda_1, \dots, \lambda_k\). For each eigenvalue, the corresponding eigenspace is the null space of \(A - \lambda I\):

\[ E_\lambda = \ker(A - \lambda I) = \{\mathbf{v} \neq \mathbf{0} : (A - \lambda I)\mathbf{v} = \mathbf{0}\}. \]

This tool:

Builds the characteristic polynomial \(\det(A - \lambda I)\).
Solves for eigenvalues (including complex ones for 2×2 matrices).
Computes a basis of eigenvectors for each eigenvalue.
Reports algebraic and geometric multiplicities.
Checks whether the matrix is diagonalizable.

Formulas for 2×2 matrices

For a 2×2 matrix

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \]

the characteristic polynomial is

\[ p(\lambda) = \det(A - \lambda I) = \det\begin{bmatrix} a-\lambda & b \\ c & d-\lambda \end{bmatrix} = (a-\lambda)(d-\lambda) - bc. \]

Expanding gives

\[ p(\lambda) = \lambda^2 - (a + d)\lambda + (ad - bc). \]

Trace: \(\operatorname{tr}(A) = a + d\).
Determinant: \(\det(A) = ad - bc\).

The eigenvalues are the roots of this quadratic:

\[ \lambda_{1,2} = \frac{\operatorname{tr}(A) \pm \sqrt{\operatorname{tr}(A)^2 - 4\det(A)}}{2}. \]

For each eigenvalue \(\lambda\), we solve \((A - \lambda I)\mathbf{v} = \mathbf{0}\) to find eigenvectors. For example, if \(b \neq 0\), one eigenvector is

\[ \mathbf{v} = \begin{bmatrix} 1 \\ \dfrac{\lambda - a}{b} \end{bmatrix}. \]

Algebraic vs geometric multiplicity

Algebraic multiplicity of \(\lambda\): how many times it appears as a root of \(p(\lambda)\).
Geometric multiplicity of \(\lambda\): dimension of its eigenspace \(E_\lambda\).

Always:

\(1 \leq \text{geometric multiplicity} \leq \text{algebraic multiplicity}\).
The sum of algebraic multiplicities equals the matrix size \(n\).

When is a matrix diagonalizable?

A matrix \(A\) is diagonalizable over \(\mathbb{C}\) if there exists an invertible matrix \(P\) such that

\[ P^{-1} A P = D, \]

where \(D\) is diagonal. This happens exactly when there are \(n\) linearly independent eigenvectors, i.e. when the sum of geometric multiplicities equals \(n\).

If \(A\) has \(n\) distinct eigenvalues, it is diagonalizable.
If some eigenvalue has geometric multiplicity < algebraic multiplicity, \(A\) is defective and not diagonalizable.

FAQ

What are eigenvalues and eigenvectors in simple terms?

Think of a matrix as a linear transformation that stretches, rotates, or shears space. An eigenvector is a direction that the transformation does not rotate—only stretches or flips. The factor by which it is stretched is the eigenvalue.

Can a matrix have complex eigenvalues?

Yes. Real matrices can have complex eigenvalues, which always come in conjugate pairs (e.g. \(a + bi\) and \(a - bi\)). A classic example is a pure rotation matrix in 2D, which has no real eigenvectors but has complex eigenvalues \(e^{\pm i\theta}\).

What if the matrix has repeated eigenvalues?

Repeated eigenvalues are not a problem by themselves. The key question is whether you still get enough linearly independent eigenvectors:

If geometric multiplicity equals algebraic multiplicity for each eigenvalue, the matrix is diagonalizable.
If not, the matrix is defective and cannot be diagonalized (though it has a Jordan normal form).

How accurate is this calculator?

For 2×2 matrices, eigenvalues are computed exactly using the quadratic formula and simplified to a + bi form when necessary. For 3×3 matrices, a numeric eigenvalue solver is used with double‑precision arithmetic. Very ill‑conditioned matrices may show small numerical errors (e.g. 1.0000000002 instead of 1).