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.^[1][2]

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.^[2]

The abstraction combines an exact algebraic test, a representation-change interpretation, and a classification purpose. It is not visual or numerical resemblance.

Structural Signature

The recognition roles are:

1. Common field: entries of both matrices and the change matrix belong to a declared field \(F\).
2. Equal square size: \(A\) and \(B\) both lie in \(M_n(F)\).
3. Invertible conjugator: \(P\in GL_n(F)\).
4. Conjugation equation: \(B=P^{-1}AP\), under a consistent coordinate convention.
5. Two bases: columns of \(P\), or its inverse depending on convention, encode the basis conversion.
6. One operator: the two matrices represent the same abstract endomorphism after identifying coordinate spaces.
7. Equivalence class: reflexivity, symmetry, and transitivity partition square matrices into similarity classes.
8. Preserved invariants: operator invariants agree across the class.
9. Non-invariant coordinates: entries and coordinate eigenvectors can differ.
10. Canonical representative: Jordan or rational canonical form can simplify classification when its hypotheses hold.

The invariant is exact: same-size square matrices related by conjugation with an invertible matrix over the declared field.

What It Is Not

It is not equality: equal matrices are similar via the identity, but similar matrices can have different entries.

It is not matrix equivalence \(B=PAQ\) with two independently chosen invertible matrices. That relation classifies rank and applies to rectangular matrices; similarity uses the same basis change on input and output.

It is not congruence \(B=P^{T}AP\) or \(P^*AP\), used for bilinear, quadratic, or sesquilinear forms.

It is not unitary similarity, orthogonal similarity, or permutation similarity. Those require the conjugator to lie in a restricted subgroup and therefore are finer relations.

It is not equality of characteristic polynomials, eigenvalues, trace, determinant, or rank. These are necessary invariants but individually and jointly in common partial lists need not be complete.

It is not geometric similarity, a similarity matrix used in data analysis, or approximate numerical closeness.

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.^[3] The definition also requires square matrices because one basis change must govern both the domain and codomain of an endomorphism.

Infinite-dimensional operator equivalence and numerical approximate similarity require additional topological or analytic structure and are outside this exact finite-matrix node.

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.

Always state the field and any restriction on \(P\). Conjugacy inside a subgroup \(H<GL_n(F)\) can distinguish matrices that are similar in the full general linear group.

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.

The relation also prevents false distinctions. Different tables of numbers may encode the same transformation, while matrices sharing easy invariants may still belong to different classes.

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 \(f\),

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

Therefore polynomial identities, ranks of polynomial expressions, characteristic and minimal polynomials, determinant, trace, and nilpotent structure are preserved. Conversely, complete invariant systems such as rational canonical form decide orbit equality.

If \(v\) is an eigenvector of \(A\), its coordinate transform is an eigenvector of \(B\). The abstract eigenspace is preserved although its coordinate subset changes.

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.

Similarity becomes analogy if two datasets, images, or organizations merely “look similar” without the conjugation equation.

Examples

Basis permutation. For \(A=\operatorname{diag}(1,2)\) and the coordinate-swap matrix \(P\), \(P^{-1}AP=\operatorname{diag}(2,1)\). Entries move but the operator is unchanged.

Diagonalization. A matrix with a basis of eigenvectors is similar to a diagonal matrix. The eigenvector matrix provides \(P\).

Rotation coordinates. A three-dimensional rotation expressed in a basis aligned with its axis has a simple block form. Conjugating by the basis change yields its matrix in the original coordinates.

Jordan classification. Two complex matrices with the same eigenvalues but different Jordan block sizes are not similar.

Negative—same characteristic polynomial. The zero \(2\times2\) matrix and \(\begin{bmatrix}0&1\\0&0\end{bmatrix}\) both have \(t^2\), but ranks 0 and 1; they are not similar.

Negative—matrix equivalence. Any two same-size matrices of equal rank are equivalent under independent row and column basis changes, but need not be similar.

Structural Tensions

T1: Coordinate variability versus operator identity. Entries change while basis-independent action remains.

T2: Simple invariants versus complete classification. Trace and eigenvalues are cheap filters; canonical data may be required.

T3: Simple representative versus expensive transformation. Canonical form clarifies structure, but computing it can be numerically unstable or symbolically costly.

T4: Full-group similarity versus restricted conjugacy. An unrestricted basis change may be inadmissible when orthogonality, unitarity, or combinatorial labels must be preserved.

T5: Algebraic exactness versus numerical computation. Exact orbit equality differs from approximate floating-point resemblance.

Structural–Framed Character

Matrix Similarity is strongly structural. Over a declared field, the existence of an invertible conjugator is an exact condition, and the resulting relation has formal equivalence and invariant properties.

The frame enters through field, admissible subgroup, exact versus numerical arithmetic, and the canonical form chosen for analysis. These parameters modify tests or refinements without weakening the central conjugation identity.

Structural Core vs. Domain Accent

The structural core is orbit equivalence under a reversible group action: representations connected by an allowed transformation count as the same, and invariant properties descend to the orbit.

The domain accent consists of matrices over a field, invertible change of basis, conjugation, linear operators, characteristic and minimal polynomials, and canonical forms. Removing it yields generic Equivalence Relation rather than Matrix Similarity.

Instantiates / Related Primes

The minimal prospective placement is a strict subsumption/specializes edge to live prime:equivalence_relation. Matrix Similarity is itself an equivalence relation on \(M_n(F)\), with an exact domain and criterion. It is therefore a mathematical specialization, not merely a user of equivalence.

Symmetry, invariance, transformation, and representation are related. prime:conjugate_variables and domain_specific:coxeter_element are false semantic coverage: neither defines the matrix orbit relation.

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

Not to Be Confused With

Matrix equivalence: independent invertible left and right multiplication.

Matrix congruence: \(P^TAP\) or \(P^*AP\), preserving form structure.

Unitary or orthogonal similarity: restricted-conjugator refinements.

Permutation similarity: relabeling by a permutation matrix.

Same eigenvalues: necessary but generally insufficient.

Conjugate variables: a different concept in physics and mathematical modeling.

Geometric similarity or similarity matrix: unrelated uses of “similarity.”

References

[1] Axler, Sheldon. Linear Algebra Done Right, 4^th ed. Springer, 2024. Operators, matrices under basis change, minimal polynomials, and canonical structure. https://linear.axler.net/. registry ↩

[2] Horn, Roger A., and Charles R. Johnson. Matrix Analysis, 2^nd ed. Cambridge University Press, 2012. Similarity, invariants, and canonical structures. https://doi.org/10.1017/CBO9781139020411. registry ↩a ↩b

[3] Roman, Steven. Advanced Linear Algebra, 3^rd ed. Springer, 2008. Rational and Jordan canonical forms and similarity classification. https://doi.org/10.1007/978-0-387-72831-5. registry ↩