`eig` — Compute eigenvalue decompositions in MATLAB and RunMat.

eig(A) computes eigenvalues of square matrix A; eig(A,B) computes generalized eigenvalues for dense square matrix pencils. Additional outputs return right and left eigenvectors under MATLAB and RunMat multi-output semantics.

Syntax

d = eig(A)
d = eig(A, options...)
[V, D] = eig(A)
[V, D] = eig(A, options...)
[V, D, W] = eig(A)
[V, D, W] = eig(A, options...)

All supported eig forms

d = eig(A)
d = eig(A, options...)
[V, D] = eig(A)
[V, D] = eig(A, options...)
[V, D, W] = eig(A)
[V, D, W] = eig(A, options...)
d = eig(A, B)
d = eig(A, B, options...)
[V, D] = eig(A, B)
[V, D] = eig(A, B, options...)
[V, D, W] = eig(A, B)
[V, D, W] = eig(A, B, options...)

Inputs

Name	Type	Required	Default	Description
`A`	`NumericArray`	Yes	—	Input square matrix.
`options`	`Any`	Variadic	—	Optional selectors (`balance`, `nobalance`, `vector`, `matrix`).
`B`	`NumericArray`	Yes	—	Second square matrix in the generalized problem AV = BV*D.

Returns

Name	Type	Description
`d`	`NumericArray`	Eigenvalues as a column vector.
`V`	`NumericArray`	Right eigenvectors.
`D`	`NumericArray`	Diagonal matrix (or vector when `vector` option is used).
`W`	`NumericArray`	Left eigenvectors.

Returned values from eig depend on how many outputs the caller requests.

Errors

Identifier	When	Message
`RunMat:eig:InvalidArgument`	Option arguments or requested output count are invalid.	`eig currently supports at most three outputs`
`RunMat:eig:InvalidInput`	Input is unsupported or matrix shape is invalid.	`eig: input matrix must be square`
`RunMat:eig:Internal`	Runtime cannot compute or materialize eig outputs.	`eig: internal runtime failure`

How `eig` works

Single output d = eig(A) returns the eigenvalues as an n × 1 column vector (values may be complex even when A is real).
Two outputs [V,D] = eig(A) return the eigenvectors (columns of V) and the diagonal eigenvalue matrix D.
Three outputs [V,D,W] = eig(A) additionally return the left eigenvectors W satisfying W' * A = D * W'.
Generalized calls d = eig(A,B), [V,D] = eig(A,B), and [V,D,W] = eig(A,B) solve A*V = B*V*D for square, same-size numeric or logical matrices. Native BLAS/LAPACK builds use ZGGEV and can return infinite eigenvalues for singular pencils; non-LAPACK builds support the nonsingular B case.
The selector 'vector' may be supplied (eig(A,'vector'), eig(A,B,'vector'), [V,d] = eig(A,'vector')) to request the eigenvalues as a column vector even when two outputs are requested. 'matrix' resets the second output to the diagonal-matrix form.
Logical and integer inputs are promoted to double precision. Complex inputs remain complex throughout the factorisation.
Empty and scalar matrices follow MATLAB’s shape conventions (eig([]) returns [], eig(5) returns 5).
When the BLAS/LAPACK feature is unavailable and B is singular, RunMat detects that the fallback path would need full QZ support and raises a clear error instead of returning a partial factorization.
Optional balancing keywords ('balance', 'nobalance') are accepted. Balancing defaults to on; the current implementation leaves balancing as a no-op while retaining the option for forward compatibility.

Does RunMat run `eig` on the GPU?

The WGPU provider implements the reserved standard eig(A) hook by downloading the input, running the same CPU decomposition used by the host path, and immediately re-uploading the double-precision eigenvalues, eigenvectors, and diagonal matrix. When the spectrum is real, the outputs therefore remain on the GPU without any user intervention.

If any output requires complex storage, you pass generalized eig(A,B), or you pass 'nobalance', RunMat automatically falls back to the host implementation and returns CPU-resident arrays. The 'vector' selector also triggers this transparent host fallback today. Reapply gpuArray if you want to continue on the device after such a fallback.

Because eig is a residency sink, the fusion planner treats it as a barrier and does not attempt to fuse surrounding elementwise work.

Examples

Computing Eigenvalues of a 2x2 Matrix

A = [2 1; 0 3];
d = eig(A)

Expected output:

d = [2; 3]

Diagonalizing a Matrix with Two Outputs

A = [0 1; -2 -3];
[V,D] = eig(A);
recon = V * D / V

Retrieving Left Eigenvectors with Three Outputs

A = [4 2; 1 3];
[V,D,W] = eig(A);
check = W' * A - D * W'

Eigenvalues of a Complex-Valued Matrix

A = [1+2i, 2-1i; 0, -3i];
[V,D] = eig(A);
diag(D)      % Complex eigenvalues

Eigenvalues of a Diagonal Matrix

A = diag([10, -2, 7]);
d = eig(A)

Handling Repeated Eigenvalues

A = [3 1 0; 0 3 0; 0 0 5];
d = eig(A)

Using the 'nobalance' Option

A = [1e6 1; 0 1e-6];
d_balanced = eig(A);
d_nobalance = eig(A, 'nobalance')

Returning Eigenvalues as a Vector with Two Outputs

A = [0 1; -2 -3];
[V,d] = eig(A, 'vector');
size(d)    % 2 x 1 column vector

Solving a Generalized Eigenvalue Problem

A = [2 0; 0 9];
B = [1 0; 0 3];
d = eig(A, B)

Expected output:

d = [2; 3]

Generalized Eigenvectors

A = [4 1; 0 9];
B = [2 0; 0 3];
[V,D,W] = eig(A, B);
rightResidual = A*V - B*V*D;
leftResidual = W'*A - D*W'*B

Running eig on a gpuArray

G = gpuArray(randn(128));
d = eig(G);          % Real spectra stay on the GPU when the provider implements eig
isa(d, 'gpuArray')   % logical 1 when the provider kept the result on device

How RunMat validates `eig`

eig uses a dense complex Schur decomposition for standard problems. Generalized eig(A,B) uses LAPACK ZGGEV when the native BLAS/LAPACK feature is enabled, including singular pencils with infinite eigenvalues; otherwise it solves the equivalent standard problem B\A for nonsingular B and rejects singular fallback cases with a descriptive QZ-required error. CPU correctness is exercised by the in-module tests linked below; GPU residency is validated at the host-fallback boundary described above rather than via a separate GPU eigensolver.

Implementation: crates/runmat-runtime/src/builtins/math/linalg/factor/eig.rs
Parity test: eig unit tests
Tolerance: CPU-only validated; no dedicated GPU eigensolver

See Correctness & Trust for the full methodology and coverage table.

Using `eig` with coding agents

Open a RunMat example with live inputs, then ask the agent to explain how eig changes the result.

Run a small eig example, explain the result, then change one input and compare the output.

FAQ

What shapes does eig support?⌄

eig requires a square matrix. Scalars are treated as 1×1 matrices, and empty inputs return empty outputs. Non-square inputs raise an error, matching MATLAB.

Do I always get complex outputs?⌄

Eigenvalues and eigenvectors are returned as complex arrays when necessary. If all imaginary parts are numerically zero, RunMat returns real doubles for convenience, mirroring MATLAB behaviour.

How do I obtain the eigenvalues as a vector when requesting eigenvectors?⌄

Pass the 'vector' selector. For example, [V,d] = eig(A,'vector') returns a column vector d and the right eigenvectors in V. Use 'matrix' (or omit the selector) when you prefer the diagonal-matrix form.

What about the optional balancing keywords?⌄

'balance' (default) and 'nobalance' are accepted. The current release treats balancing as a no-op; the option remains so future releases can introduce a true balancing implementation without breaking user code.

Are generalized eigenvalue problems supported?⌄

Yes. In native BLAS/LAPACK builds, eig(A,B) uses LAPACK ZGGEV for dense generalized matrix pencils and can return infinite eigenvalues when the pencil is singular. In non-LAPACK builds, RunMat supports the nonsingular B case by solving the equivalent standard problem B\A and raises a descriptive QZ-required error for singular B.

How are left eigenvectors normalised?⌄

RunMat scales standard left eigenvectors so that W' * V = I. For generalized eig(A,B), it scales them so that W' * B * V = I. When a provider supplies the GPU implementation the same normalisation is expected.

Does eig participate in fusion or auto-offload?⌄

No. Eigenvalue decomposition executes eagerly and acts as a residency sink. The fusion planner will gather any GPU-resident inputs before factorisation.

How can I continue on the GPU after calling eig today?⌄

When the active provider implements the standard eig(A) hook (the WGPU backend does for real spectra), the outputs remain on the GPU automatically. If the decomposition falls back to the CPU—because the call is generalized, the spectrum is complex, or 'nobalance' was requested—call gpuArray on whichever results you need on the device.

What happens if the eigenvector matrix is singular?⌄

When the right eigenvectors form a singular matrix, RunMat falls back to computing left eigenvectors from the conjugate-transposed problem. If that fails, requesting the third output raises an error, matching MATLAB’s failure behaviour.

What equation does eig solve?⌄

For the standard form [V, D] = eig(A), the outputs satisfy A*V = V*D, where the columns of V are right eigenvectors and D is a diagonal matrix of eigenvalues. For the generalized form [V, D] = eig(A, B), the outputs satisfy A*V = B*V*D when B is nonsingular.

How do I get just the eigenvalues versus eigenvalues and eigenvectors?⌄

Call d = eig(A) with a single output to get a column vector of eigenvalues only. Use [V, D] = eig(A) to also get the eigenvectors, or [V, D, W] = eig(A) to additionally get the left eigenvectors in W.

Are the eigenvalues returned by eig sorted?⌄

No. Unlike svd, which returns singular values in descending order, eig returns eigenvalues in whatever order the underlying decomposition produces. Sort them yourself when you need ordering, for example [d, idx] = sort(diag(D)); D = diag(d); V = V(:, idx);.

When should I use eig versus eigs?⌄

Use eig for dense matrices when you want the full spectrum. Use eigs for sparse or very large matrices when you only need a few eigenvalues (for example the k largest or smallest), since it uses iterative Arnoldi/Lanczos methods instead of a full factorization.

Factor

chol · lu · qr · svd

Open-source implementation

Unlike proprietary runtimes, every RunMat function is open-source. Read exactly how eig is executed, line by line, in Rust.

View the source for eig in Rust on GitHub
Learn how the RunMat runtime works
Found a bug? Open an issue with a minimal reproduction.

About RunMat

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eig — Compute eigenvalue decompositions in MATLAB and RunMat.