Adjacency Matrix Eigenvalues Calculator

Enter matrix values

Use the responsive calculator grid below. Large screens show three columns, smaller screens show two, and mobile shows one.

Matrix size

Graph interpretation

Quick example

Matrix editor

Example data table

This sample uses a simple four-node path graph. The spectrum is fully real because the adjacency matrix is symmetric.

Example	Matrix	Eigenvalues	Spectral Radius	Energy
4-node path graph	[0 1 0 0; 1 0 1 0; 0 1 0 1; 0 0 1 0]	1.618034, 0.618034, -0.618034, -1.618034	1.618034	4.472136

Formula used

Characteristic equation:
det(A − λI) = 0

Spectral radius:
ρ(A) = max |λᵢ|

Graph energy:
E(A) = Σ |λᵢ|

Trace and determinant checks:
trace(A) = Σ λᵢ and det(A) = Π λᵢ

The file estimates eigenvalues numerically from the characteristic polynomial. For symmetric adjacency matrices, the returned values should be real apart from tiny rounding noise.

How to use this calculator

Choose a square matrix size from 2 × 2 up to 6 × 6.
Select auto, directed, or undirected interpretation.
Enter each adjacency value in the matrix editor.
Use the helper buttons to zero, mirror, or preload values.
Press Calculate Eigenvalues to display results above the form.
Review eigenvalues, radius, energy, degree-style summaries, and the Plotly complex-plane graph.
Export the current result set through the CSV or PDF buttons.

Frequently asked questions

1. What does an adjacency matrix eigenvalue represent?

Each eigenvalue captures a structural mode of the graph. Large magnitudes often indicate stronger connectivity patterns, repeated values can reflect symmetry, and zero values may suggest rank deficiencies or disconnected structure.

2. Why can some eigenvalues be complex?

Complex eigenvalues commonly appear when the matrix is not symmetric. Directed graphs and general weighted matrices can create conjugate pairs, which is why the calculator plots the spectrum on the complex plane.

3. What is the spectral radius used for?

The spectral radius is the largest eigenvalue magnitude. It is useful in stability checks, diffusion analysis, growth behavior, and comparing how strongly different adjacency structures can propagate influence.

4. Why compare trace and determinant with eigenvalues?

These provide quick consistency checks. For exact arithmetic, the trace equals the sum of eigenvalues, and the determinant equals their product. Small differences usually come from numerical rounding.

5. Does the tool support weighted graphs?

Yes. Entries do not need to be limited to zeros and ones. You can enter any real weights, making the calculator useful for weighted networks, transition-style matrices, and generalized graph models.

6. Why are symmetric matrices easier to interpret?

Symmetric adjacency matrices correspond to undirected graphs. Their eigenvalues are real, which simplifies ordering, plotting, and interpreting spectral measures such as energy, dominant modes, and multiplicities.

7. What does graph energy mean here?

Graph energy is the sum of eigenvalue magnitudes. It is a compact way to summarize the overall spectral spread of the adjacency matrix and compare how differently two graphs are organized.

8. Is this calculator suitable for large matrices?

This file is optimized for compact educational and analytical cases up to 6 × 6. Larger matrices usually require specialized numerical libraries for speed, conditioning, and more advanced decomposition control.