Matrix Similarity

Treat square matrices A and B over the same field as equivalent exactly when B = P⁻¹AP for an invertible P, so they represent one linear operator in different bases.

Version

v1 · 2026-08-30 · History

Domain-specific #

2245

Origin domain

linear algebra

Subdomain

matrix canonical-form theory

Aliases

Similar matrices, Similarity of matrices, Matrix conjugacy, Similarity transformation

Core Idea

Two square matrices \(A,B\in M_n(F)\) are similar when an invertible matrix \(P\in GL_n(F)\) satisfies

\[ B=P^{-1}AP. \]

This is precisely the change-of-basis relation: \(A\) and \(B\) are coordinate matrices of the same linear operator on an \(n\)-dimensional vector space, expressed in two ordered bases.

Similarity groups all basis-dependent representations of an operator into one equivalence class. It preserves basis-independent operator properties—characteristic and minimal polynomials, determinant, trace, rank, eigenvalue multiplicities, and canonical-form data—while allowing entries, eigenvectors in coordinates, sparsity, and appearance to change.

Scope of Application

Matrix Similarity applies throughout finite-dimensional linear algebra and matrix theory. It supports canonical forms, diagonalization, spectral analysis, linear recurrences, differential equations, dynamical systems, control, representation theory, and any setting where an endomorphism's coordinate matrix changes with basis.

The base field matters. Jordan form may require an algebraically closed field or polynomial splitting, whereas rational canonical form exists over any field. The definition also requires square matrices because one basis change must govern both the domain and codomain of an endomorphism.

Clarity

To prove similarity construct an invertible \(P\) and verify \(AP=PB\), equivalently \(B=P^{-1}AP\). The intertwining form avoids explicitly computing an inverse.

To disprove similarity, find a similarity invariant that differs: characteristic polynomial, minimal polynomial, rank of \((A-\lambda I)^k\), elementary divisors, or canonical form. Equal eigenvalues alone are insufficient. For example, the \(2\times2\) zero matrix and the nonzero nilpotent Jordan block both have characteristic polynomial \(t^2\) but different ranks and minimal polynomials.

Manages Complexity

A single operator has infinitely many coordinate matrices as bases vary. Similarity quotients this representational proliferation into one class, letting reasoning operate on invariant structure rather than arbitrary coordinates.

Canonical forms make the compression computational. A complicated matrix can be replaced by a similar diagonal, Jordan, or rational canonical representative where powers, exponentials, invariant subspaces, and recurrence behavior are easier to analyze.

Abstract Reasoning

The group \(GL_n(F)\) acts on \(M_n(F)\) by conjugation:

\[ P\cdot A=P^{-1}AP. \]

Similarity classes are the orbits of this action. Reflexivity uses \(I\); symmetry replaces \(P\) by \(P^{-1}\); transitivity composes conjugators.

For every polynomial \(f\),

\[ f(P^{-1}AP)=P^{-1}f(A)P. \]

Knowledge Transfer

Literal transfer occurs wherever square matrices encode one linear operator under different bases. The formula and invariant logic remain unchanged across scientific applications.

The portable residue is an equivalence relation induced by reversible representation change. Live prime:equivalence_relation supplies partition, representative choice, and invariant reasoning. Matrix Similarity adds fields, square matrices, invertible conjugation, basis coordinates, and canonical forms.

Relationships to Other Abstractions

Current abstraction Matrix Similarity Domain-specific

Parents (1) — more general patterns this builds on

• Matrix Similarity is a kind of Equivalence Relation Prime

  The minimal prospective placement is a strict subsumption/specializes edge to live prime:equivalence_relation.

Hierarchy path (1) — routes to 1 parentless root

• Matrix Similarity → Equivalence Relation

Neighborhood in Abstraction Space

Matrix Similarity sits in a sparse region of the domain-specific corpus (76^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Applied Linear & Special Functions (18 abstractions)

Nearest neighbors

• Quadratic Space — 0.86
• Pseudo-Euclidean Space — 0.83
• Categorical Lift — 0.83
• Quadratic Field — 0.83
• Roothaan–Hall Equations — 0.82

Computed from structural-signature embeddings · 2026-09-08