Eigenvalue in MATLAB: eig, eigs and Eigenvectors

Short answer: in MATLAB, e = eig(A) returns the eigenvalues of a square matrix A as a column vector, and [V,D] = eig(A) returns the eigenvectors in the columns of V and the eigenvalues on the diagonal of D, so that A*V = V*D. For large sparse matrices where you need only a few eigenvalues, use eigs instead. This guide is for researchers and analysts who need to compute, check and interpret eigenvalues in MATLAB; it follows the MathWorks reference pages for eig and eigs (read 2026-11-07).

What eigenvalues are used for in research

Eigenvalues and eigenvectors appear wherever a linear system is analysed: principal components of a covariance matrix, vibration modes and stability of dynamical systems, and Markov or population-matrix models. This page covers the MATLAB mechanics (which function, which call form, how to read and verify the output); it does not teach the underlying linear algebra.

Install or access (official source only)

eig and eigs are documented in the MATLAB reference at mathworks.com/help/matlab. Obtain and activate MATLAB through MathWorks as described in the MATLAB installation guide. If you also need to build the matrix first, see matrices in MATLAB.

Step by step: eigenvalues in MATLAB

1. Eigenvalues only: matlab eigenvalues with eig(A)

1. Create a square matrix, for example A = gallery("lehmer",4);
2. Run e = eig(A). With one output, MATLAB returns a column vector. For this Lehmer matrix the documentation lists eigenvalues of about 0.2078, 0.4078, 0.8482 and 2.5362.

2. Eigenvectors and eigenvalues in MATLAB: [V,D] = eig(A)

3. Run [V,D] = eig(A). With two outputs, D is a diagonal matrix of eigenvalues and each column of V is the matching right eigenvector.
4. Optional: [V,D,W] = eig(A) also returns left eigenvectors in W, satisfying W'*A = D*W'.
5. Optional: add "vector" or "matrix" as the last argument to choose the output form of the eigenvalues ([V,d] = eig(A,"vector")).

3. Generalized problem: eig(A,B)

For A*v = λ*B*v, call e = eig(A,B) or [V,D] = eig(A,B). The algorithm argument selects "chol" (default for the symmetric case) or "qz". When B is singular, eig(A,B) still returns a result: the documentation’s example with A = eye(2) returns one finite eigenvalue (0.0909) and one Inf.

4. Large sparse matrices: eigs

d = eigs(A) returns the six largest-magnitude eigenvalues; eigs(A,k) returns k of them; eigs(A,k,sigma) targets eigenvalues near sigma ("smallestabs" is the same as sigma = 0). [V,D,flag] = eigs(___) adds a convergence flag, where 0 means all requested eigenvalues converged. For a function handle, use eigs(Afun,n,___).

Which function: eig, eig(A,B) or eigs

Situation	Call	Source note
All eigenvalues of a standard problem	eig(A)	Returns all eigenvalues
Generalized problem, including singular B	eig(A,B)	Works when B is singular
Sparse, real and symmetric, all eigenvalues	eig(A)	Documented as supported
Sparse eigenvectors, or non-symmetric sparse eigenvalues	eigs	The eig page directs these cases to eigs
A few eigenvalues of a large matrix	eigs(A,k)	Useful when computing all of them is expensive; for small dense matrices eig(full(A)) may be quicker

Output and how to read it

• Check the residual. For the circulant matrix A = gallery("circul",3), the documentation reports eigenvalues 6 and a complex pair near −1.5 ± 0.866i, and a residual A*V - V*D on the order of 1e-14. That is floating-point round-off, not an error.
• Eigenvalues are not sorted. For magic(5), D is unsorted. Sort and reorder together: [d,ind] = sort(diag(D)); Ds = D(ind,ind); Vs = V(:,ind);
• Special matrices. Real symmetric or complex Hermitian input gives real eigenvalues; skew-symmetric or skew-Hermitian input gives purely imaginary ones.
• Eigenvector scaling. Each eigenvector has 2-norm 1 and, for symmetric, Hermitian or skew-Hermitian A, the vectors are orthonormal. With "nobalance" the norms are not necessarily 1.
• Reproducibility. Signs and complex phases of eigenvectors can differ between machines and MATLAB releases, so compare A*V - V*D rather than raw eigenvector entries.

For related matrix operations, see matrix multiplication in MATLAB and solving equations in MATLAB.

Troubleshooting

• Error that the matrix must be square: eig takes a square matrix; check size(A).
• Eigenvalues appear in a different order than a textbook or another tool: expected, since eig output is unordered; sort as shown above.
• Tiny nonzero imaginary parts or residuals around 1e-14: round-off, as in the circulant example.
• Sparse input and eigenvectors requested: use eigs, as the eig page directs.
• Different results with "nobalance": balancing is on by default; turning it off can change eigenvector norms.

Free alternatives

Tool	Call	Documented behaviour
NumPy	numpy.linalg.eig(a)	Returns eigenvalues and unit-length eigenvectors (columns); eigenvalues not necessarily ordered; docs recommend eigh for symmetric or Hermitian input
GNU Octave	[V, lambda] = eig(A)	Same call forms as MATLAB, including eig(A, B); eigenvalues are not ordered
R (base)	eigen(x)	Returns a list with values and vectors; values sorted by decreasing modulus for asymmetric matrices; vectors normalized to unit length

Note the difference: R sorts for asymmetric matrices, while MATLAB, NumPy and Octave do not guarantee any order.

FAQ

How do I find eigenvalues in MATLAB?

Use e = eig(A) for a square matrix A; the result is a column vector of eigenvalues.

How do I get eigenvectors as well as eigenvalues in MATLAB?

Use [V,D] = eig(A). Columns of V are eigenvectors and the diagonal of D holds the eigenvalues, with A*V = V*D.

Does eig sort eigenvalues?

No. The documentation’s magic(5) example returns D unsorted; sort diag(D) and reorder the columns of V to match.

When should I use eigs instead of eig?

For large sparse matrices when you need only some eigenvalues, or for eigenvectors of sparse matrices and eigenvalues of non-symmetric sparse matrices.

Sources

• MathWorks, eig: Eigenvalues and eigenvectors, read 2026-11-07.
• MathWorks, eigs: Subset of eigenvalues and eigenvectors, read 2026-11-07.
• NumPy, numpy.linalg.eig, read 2026-11-07.
• GNU Octave documentation, Basic Matrix Functions, read 2026-11-07.
• R documentation, eigen, read 2026-11-07.