Symbolic Matrix Analyzer

Characteristic polynomial, spectrum and exact diagonalization

Input Matrix

Matrix Dimension

Preset

Global Scalar Factor

Symbols

Results

Enter a symbolic or numeric matrix in the panel
and press Analyze to view the step-by-step development.

Fundamentals & Explanation

Matrices and Linear Transformations

A square matrix of order n is an n × n array of numbers (or symbolic expressions) that represents a linear transformation on a vector space. Each column indicates where the corresponding canonical basis vector gets mapped to.

A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, \quad A\mathbf{v} = \mathbf{w}

Analytical Rigor: This analyzer uses exact symbolic algebra (SymPy engine), not floating-point numerical approximations. Eigenvalues and eigenvectors are expressed in closed form — fractions, radicals, and complex numbers included — with zero rounding error.

Determinant and Trace

The determinant encodes the volumetric behavior of the transformation, while the trace (sum of the diagonal) equals the sum of all eigenvalues. Both are invariant under change of basis.

2×2 Determinant

\det(A) = a_{11}a_{22} - a_{12}a_{21}

Trace

\operatorname{tr}(A) = \sum_{i=1}^{n} a_{ii} = \sum_{i=1}^{n} \lambda_i

det(A) = 0: The transformation collapses the space — singular matrix, not invertible.
det(A) < 0: Orientation is reversed (mirror-like reflection).
|det(A)|: Scale factor for the area (2D) or volume (3D) of the transformed space.

Characteristic Polynomial and Spectrum

Eigenvalues are the scalars λ for which the transformation (A − λI) collapses the space (det = 0). They are the roots of the degree-n characteristic polynomial:

p(\lambda) = \det(A - \lambda I) = 0

The full set of eigenvalues is called the spectrum of the matrix. By the Cayley-Hamilton Theorem, every matrix satisfies its own characteristic polynomial: p(A) = 0.

Eigenvectors and Multiplicities

An eigenvector v associated with eigenvalue λ is a nonzero vector that the transformation only scales — never rotates:

A\mathbf{v} = \lambda\mathbf{v} \iff (A - \lambda I)\mathbf{v} = \mathbf{0}

Algebraic multiplicity: how many times λ appears as a root of p(λ).
Geometric multiplicity: dimension of the null space of (A − λI) — number of independent eigenvectors.
Complex eigenvalues: appear in conjugate pairs and indicate a rotational component.
Defective matrix: when geometric multiplicity is strictly less than algebraic multiplicity.

Diagonalization

A matrix is diagonalizable if and only if it has n linearly independent eigenvectors. In that case the spectral decomposition exists:

A = P \cdot D \cdot P^{-1}

where P holds the eigenvectors as columns and D is diagonal with the eigenvalues. This enables computing matrix powers in linear time:

A^n = P \cdot D^n \cdot P^{-1}

Matrices with repeated eigenvalues may or may not be diagonalizable. When they are not, their canonical form is the Jordan normal form, with Jordan blocks along the diagonal.

Real-World Applications

Google PageRank: Models the web as a giant stochastic matrix and computes its dominant eigenvector (eigenvalue λ = 1).
PCA (Principal Component Analysis): Decomposes the covariance matrix; eigenvectors define the directions of maximum variance.
Quantum Mechanics: Observables are Hermitian operators; their eigenvalues are the only possible outcomes of a measurement.
Special Relativity: The Lorentz boost matrix has a spectrum containing the relativistic Doppler factors γ(1 ± β).
Dynamical Systems:Equilibrium stability is determined by the sign of the real parts of the Jacobian's eigenvalues.

📐 Notable Cases

Identity: λ = 1 (×n)

90° Rotation: λ = ±i

Symmetric: λ ∈ ℝ always

Nilpotent: λ = 0 (×n)

⚡ Pauli Matrices

σ₁: λ = ±1

σ₂: λ = ±1

σ₃: λ = ±1

Basis of su(2) algebra

Academic References

[1] Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press.
[2] Axler, S. (2015). Linear Algebra Done Right (3rd ed.). Springer.
[3] Horn, R. A., & Johnson, C. R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press.
[4] Meurer, A. et al. (2017). SymPy: symbolic computing in Python. PeerJ Computer Science, 3, e103.

Symbolic Matrix Analyzer

Input Matrix

Results

Fundamentals & Explanation

Matrices and Linear Transformations

Determinant and Trace

Characteristic Polynomial and Spectrum

Eigenvectors and Multiplicities

Diagonalization

Real-World Applications

📐 Notable Cases

⚡ Pauli Matrices

Academic References

You may also like:

Matrices and Linear Transformations

Determinant and Trace

Characteristic Polynomial and Spectrum

Eigenvectors and Multiplicities

Diagonalization

Real-World Applications

📐 Notable Cases

⚡ Pauli Matrices

Academic References