SymbolicDiagonalization.jl

Symbolic eigenvalue computation for structured matrices

The Problem

The Abel-Ruffini theorem proves no closed-form solution exists for polynomials of degree 5+. This means general symbolic 5×5+ matrices cannot be diagonalized. However, many matrices have exploitable structure.

The Solution

This package automatically detects matrix structure and returns closed-form symbolic eigenvalues:

using Symbolics, SymbolicDiagonalization, LinearAlgebra

# 1024×1024 nested Kronecker product - degree 1024 polynomial!
# But structure reduces it to 10 quadratic problems
@variables a1 b1 c1  a2 b2 c2  a3 b3 c3  a4 b4 c4  a5 b5 c5  a6 b6 c6  a7 b7 c7  a8 b8 c8  a9 b9 c9  a10 b10 c10

matrices = [[a1 b1; b1 c1], [a2 b2; b2 c2], [a3 b3; b3 c3], [a4 b4; b4 c4], [a5 b5; b5 c5],
            [a6 b6; b6 c6], [a7 b7; b7 c7], [a8 b8; b8 c8], [a9 b9; b9 c9], [a10 b10; b10 c10]]
K = reduce(kron, matrices)  # 1024×1024 with 30 symbolic parameters

eigvals(K)  # 1024 symbolic eigenvalues in ~33 seconds

Supported Patterns

Pattern	Size	Method
Any matrix	≤4×4	Quadratic/Cardano/Ferrari formulas
Block-diagonal	Any	Recursive decomposition
Circulant	Any	DFT diagonalization
Kronecker A⊗B	Any	λ(A)·λ(B) factorization
Nested Kronecker	Any	Recursive factorization
Symmetric Toeplitz tridiagonal	Any	Chebyshev formula
Hadamard	2ⁿ	±√(2ⁿ)
DFT	Any	Fourth roots of n
Permutation	Any	Cycle decomposition
SO(2), SO(3), SO(4)	2,3,4	Rotation angle extraction
SU(2), SU(3)	2,3	Lie algebra structure
Cartan type Aₙ	Any	Root system formula
Path/Cycle Laplacian	Any	Trigonometric formula

Quick Start

using Symbolics, SymbolicDiagonalization, LinearAlgebra

@variables a b c d

Block-diagonal (recursive decomposition):

M = [a b 0 0; b a 0 0; 0 0 c d; 0 0 d c]
eigvals(M)  # [a+b, a-b, c+d, c-d]

Circulant (DFT formula works for any n):

C = [a b c; c a b; b c a]
eigvals(C)  # Closed-form for any size

Kronecker product:

A = [a 0; 0 b]
B = [c 0; 0 d]
eigvals(kron(A, B))  # [ac, ad, bc, bd]

Rotation matrices:

@variables θ φ
eigvals(kron(SO2_rotation(θ), SO2_rotation(φ)))  # e^{i(±θ±φ)}

Installation

using Pkg
Pkg.add("SymbolicDiagonalization")

Documentation

Full documentation

• User Guide - Examples and workflows
• Pattern Library - All supported patterns
• Mathematical Background - Theory

License

GPL v3 - see LICENSE.