Similarity of operators

1 Similarity of Linear Operators

1.1 Definition via conjugation

Let \(V\) and \(W\) be finite-dimensional vector spaces over a field \( \mathbb{F} \), and let \(A:V\to V\) and \(B:W\to W\) be linear operators. In the most common setting, \(V=W\). Two operators \(A\) and \(B\) are similar if there exists an invertible linear map \(P:V\to V\) such that \[ B = P^{-1} A P . \] This expresses that \(B\) is obtained from \(A\) by a change of coordinates given by \(P\) (conjugation by an invertible map). In matrix terms (once bases are chosen), if \(A\) and \(B\) represent the operators in two bases, then similarity means the representing matrices differ by conjugation by an invertible matrix.

1.2 Similarity as an equivalence relation

Similarity partitions operators into equivalence classes.

Reflexive: Taking \(P=I\), one has \(A = I^{-1} A I\), so \(A\) is similar to itself.
Symmetric: If \(B=P^{-1}AP\), then \(A = PBP^{-1}\), so \(A\) is similar to \(B\).
Transitive: If \(B=P^{-1}AP\) and \(C=Q^{-1}BQ\), then \(C=(PQ)^{-1}A(PQ)\).

Thus, similarity is an equivalence relation on the set of linear operators on a fixed vector space.

1.3 Matrix representation and change of basis

If \(A:V\to V\) is represented by a matrix \([A]_{\mathcal{B}}\) in a basis \(\mathcal{B}\), and \(\mathcal{C}\) is another basis, then the matrices are related by conjugation: \[ [A]_{\mathcal{C}} = S^{-1}[A]_{\mathcal{B}}S, \] where \(S\) is the change-of-basis matrix from \(\mathcal{B}\) to \(\mathcal{C}\). Therefore, similarity captures the extent to which the “action” of a linear map is independent of the chosen coordinates.

A useful viewpoint is that the equation \(B=P^{-1}AP\) asserts equality of operators after transporting vectors through \(P\): acting with \(A\) in the \(P\)-transformed coordinates corresponds to acting with \(B\) in the original coordinates.

1.4 Similarity in different dimensions (conditions and conventions)

Similarity is typically formulated for operators on vector spaces of the same finite dimension, because an invertible map \(P\) must exist between the underlying spaces. Common conventions include:

If \(A:V\to V\) and \(B:W\to W\), then \(A\) and \(B\) are similar only when \(\dim V=\dim W\).
Over different fields, similarity must be considered with respect to the same base field \(\mathbb{F}\), since invariants like characteristic polynomials depend on coefficient field.
In infinite-dimensional settings, defining similarity requires careful handling: an “invertible” map might mean bijective and bounded (in functional analysis) or just a linear bijection (in purely algebraic contexts). Most classification results are cleaner in finite dimensions.

2 Invariants Under Similarity

2.1 Eigenvalues and characteristic polynomials

Similarity preserves the characteristic polynomial. If \(B=P^{-1}AP\), then \[ \chi_B(t)=\chi_A(t), \] so the multiset of eigenvalues (over an algebraic closure) is unchanged. The reasoning comes from determinant identities under conjugation: matrices \(A-\lambda I\) and \(P^{-1}(A-\lambda I)P\) have the same determinant.

However, the presence of eigenvalues depends on the field: over \(\mathbb{R}\), a real matrix may have complex eigenvalues in conjugate pairs, whereas over \(\mathbb{C}\) they split completely.

2.2 Minimal polynomials

The minimal polynomial of an operator is the monic polynomial of least degree that annihilates it. Similarity preserves minimal polynomials: \[ m_B(t)=m_A(t). \] This is stronger than preserving just the characteristic polynomial, because the minimal polynomial encodes the size of Jordan blocks (in a field where Jordan form exists).

2.3 Trace and determinant

Two basic similarity invariants are:

\(\mathrm{tr}(A)\), the trace, preserved under conjugation.
\(\det(A)\), the determinant, also preserved under conjugation.

Since trace and determinant can be expressed in terms of coefficients of the characteristic polynomial, their invariance follows from characteristic polynomial invariance.

2.4 Rank, nullity, and other similarity-invariant quantities

From \(B=P^{-1}AP\), one has \[ \ker(B)=P^{-1}\ker(A) \quad \text{and} \quad \mathrm{im}(B)=P^{-1}\mathrm{im}(A), \] so \(\dim\ker(B)=\dim\ker(A)\) (nullity) and \(\dim\mathrm{im}(B)=\dim\mathrm{im}(A)\) (rank) are preserved. More generally, ranks of powers such as \(\mathrm{rank}(A^k)\) and nullities of \(A^k\) are similarity invariants, which become relevant in Jordan-theoretic classification.

2.5 Jordan structure and geometric multiplicities (overview level)

Over an algebraically closed field, similarity classes correspond to the Jordan structure: for each eigenvalue, Jordan blocks have certain sizes. Similarity preserves:

the algebraic multiplicity (from the characteristic polynomial),
the geometric multiplicity (dimension of the eigenspace),
and the arrangement of generalized eigenspaces (captured by Jordan block sizes).

Geometric multiplicities can be read from kernels of \(A-\lambda I\), and more refined data from kernels of \((A-\lambda I)^k\).

3 Canonical Forms and Classification

3.1 Jordan canonical form

The Jordan canonical form gives a complete classification of complex (or, more generally, algebraically closed-field) linear operators up to similarity. Every operator is similar to a block-diagonal matrix whose blocks are Jordan blocks: \[ J_k(\lambda)= \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\\ 0 & \lambda & 1 & \cdots & 0\\ \vdots & & \ddots & \ddots & \vdots\\ 0 & \cdots & 0 & \lambda & 1\\ 0 & \cdots & \cdots & 0 & \lambda \end{pmatrix}. \] The sizes and counts of these blocks determine the similarity class. Two operators are similar exactly when they have the same Jordan block decomposition.

3.2 Diagonalization criteria

An operator is diagonalizable if it is similar to a diagonal matrix. In Jordan terms, this happens precisely when every Jordan block has size \(1\). Equivalent criteria include:

the dimension of the eigenspace for each eigenvalue adds up to the full space dimension (a complete set of eigenvectors),
the minimal polynomial splits into distinct linear factors over the field.

Diagonalization simplifies many computations but does not apply to operators with nontrivial Jordan blocks.

3.3 Triangularization and Schur-type viewpoints (real vs complex)

Over \(\mathbb{C}\), every linear operator can be put into upper triangular form via similarity (a manifestation of triangularization results). Over \(\mathbb{R}\), one can triangularize into real blocks corresponding to real eigenvalues and \(2\times 2\) blocks corresponding to complex-conjugate eigenpairs, reflecting the field limitation.

This viewpoint emphasizes that similarity classification may be more naturally expressed through block structures when the field does not contain all eigenvalues.

3.4 Rational canonical form (conceptual overview)

Over an arbitrary field \(\mathbb{F}\), the rational canonical form replaces Jordan blocks with companion-matrix blocks determined by invariant factors. It provides a complete similarity classification without requiring algebraic closure. Conceptually:

one decomposes the operator using the structure of finitely generated \(\mathbb{F}[t]\)-modules,
invariant factors determine the form,
operators are similar if and only if they share the same invariant factors.

Rational canonical form thus generalizes Jordan form to general fields.

4 Effects on Subspaces and Spectral Data

4.1 Invariant subspaces and how similarity transforms them

If \(B=P^{-1}AP\) and \(U\subseteq V\) is \(A\)-invariant (meaning \(A(U)\subseteq U\)), then \(P^{-1}(U)\) is \(B\)-invariant: \[ B(P^{-1}U)=P^{-1}A(U)\subseteq P^{-1}U. \] Conversely, similarity carries invariant subspace structures from one operator to the other via the intertwining map \(P\).

4.2 Eigenspaces and generalized eigenspaces

For an eigenvalue \(\lambda\), eigenspaces transform naturally under similarity. If \[ \ker(A-\lambda I)=E_\lambda(A), \] then \[ E_\lambda(B)=P^{-1}E_\lambda(A). \] Likewise, generalized eigenspaces defined by \[ \ker(A-\lambda I)^k \] transform by the same rule for every \(k\). This is why Jordan block sizes, which can be read from the growth of these kernels as \(k\) increases, are similarity invariants.

4.3 Commutants: operators that commute with a given operator (motivation)

The set of operators commuting with \(A\), \[ \{X: XA=AX\}, \] is called the commutant or centralizer of \(A\). Under similarity, commutation relations are preserved: if \(B=P^{-1}AP\), then \(X\) commutes with \(A\) precisely when \(P^{-1}XP\) commutes with \(B\). This concept motivates classification efforts because the commutant reflects the internal symmetry of an operator.

4.4 Spectral mapping behavior under similarity (conceptual summary)

Similarity preserves spectral data such as eigenvalues and Jordan structure. For functional constructions, one typically uses the same polynomial or analytic functional calculus expression applied to each operator. At a conceptual level, if \(f\) is a polynomial, then \(f(B)=P^{-1}f(A)P\), so invariants of \(f(A)\) and \(f(B)\) align. In particular, eigenvalues of \(f(A)\) relate to eigenvalues of \(A\) through the mapping \( \lambda \mapsto f(\lambda)\).

5 Similarity and Matrix Conjugation Techniques

5.1 Constructing \(P\) for a given similarity claim

To prove \(B=P^{-1}AP\), one typically constructs \(P\) from information about \(A\) and \(B\). Common strategies include:

choosing a basis in which \(A\) has a known form and matching it to a basis for \(B\),
using eigenvectors or generalized eigenvectors when both operators share the same invariant data,
solving for \(P\) directly from matrix equations (effective in small dimensions).

If \(P=[p_1\ \cdots\ p_n]\) is formed from vectors, the relation \(AP=PB\) is often easier to work with: \[ AP=PB \quad \Longleftrightarrow \quad B=P^{-1}AP. \] This converts a conjugation statement into a linear system for the columns of \(P\).

5.2 Similarity of block matrices

When matrices are partitioned into blocks, similarity often respects decompositions into invariant subspaces. If \(V\) decomposes into subspaces that are invariant under \(A\), then \(A\) can be represented in a basis adapted to that decomposition, yielding block-upper-triangular structures. Similarity can then be analyzed by focusing on how blocks correspond under the conjugating map.

In practice, one may search for \(P\) that simultaneously puts the operators into comparable block forms.

5.3 Reduction via elementary operations (similarity-compatible transformations)

Row and column operations require care. The similarity relation uses conjugation \(P^{-1}AP\), which corresponds to changing basis on the domain and codomain in a coordinated way. Similarity-compatible operations include:

switching basis vectors (permutation similarity),
scaling basis vectors (diagonal similarity),
combining basis vectors in a way that corresponds to multiplying by invertible matrices on the left and right in opposite directions.

Thus, the reduction of a matrix “via elementary similarity operations” corresponds to applying an invertible transformation both to rows and columns in a synchronized manner, preserving the underlying operator up to change of coordinates.

5.4 Common pitfalls (confusing row/column operations with similarity)

A frequent mistake is treating arbitrary row operations or column operations as if they preserve similarity. In general:

Performing elementary row operations corresponds to left-multiplication by an invertible matrix, which does not preserve similarity.
Performing elementary column operations corresponds to right-multiplication, also not similarity in isolation.
Similarity corresponds to conjugation, i.e., \(A\mapsto P^{-1}AP\), not just \(A\mapsto R A\) or \(A\mapsto A C\).

Another pitfall is assuming that equal characteristic polynomials imply similarity; this is false because different Jordan structures can share the same characteristic polynomial.

6 Similarity of Operators in Special Settings

6.1 Similarity for normal/unitary/Hermitian operators (high-level)

In complex inner-product spaces, special operator classes exhibit stronger relationships between similarity, diagonalization, and orthonormal bases.

For normal operators (including unitary and Hermitian/self-adjoint operators), one has powerful spectral theorems: such operators are unitarily diagonalizable (under appropriate assumptions), which implies they are also diagonalizable by similarity. While similarity is a weaker equivalence than unitary equivalence, normality ensures the stronger outcome occurs.

6.2 Similarity vs unitary equivalence (distinction)

Similarity allows an arbitrary invertible change of basis \(P\), not necessarily preserving inner products. Unitary equivalence restricts the change of basis to unitary maps. Therefore, two operators can be similar without being unitarily equivalent. Unitarty equivalence preserves additional geometric structure (e.g., orthogonality of eigenspaces and norms).

In infinite-dimensional contexts, this distinction becomes even more significant due to boundedness and completeness issues.

6.3 Similarity in finite-dimensional vs infinite-dimensional contexts (scope)

Many classification results—especially Jordan form—are inherently finite-dimensional or rely on algebraic closure. In infinite dimensions:

conjugation by invertible operators still defines an equivalence concept,
spectral properties can differ dramatically because spectra need not consist only of eigenvalues,
canonical-form techniques may fail or require additional structure (e.g., compactness, normality, or specific operator classes).

Thus, similarity theory remains meaningful, but the finite-dimensional classification picture does not always extend directly.

7 Examples and Worked Computations

7.1 Similarity of simple \(2\times 2\) operators

Consider matrices \[ A=\begin{pmatrix}\lambda & 0\\0 & \mu\end{pmatrix},\quad B=\begin{pmatrix}\lambda & 0\\0 & \mu\end{pmatrix}. \] If \(B\) is identical to \(A\), they are trivially similar. More interestingly, let \[ A=\begin{pmatrix}\lambda & 0\\0 & \mu\end{pmatrix},\quad B=\begin{pmatrix}\mu & 0\\0 & \lambda\end{pmatrix}. \] Taking \[ P=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \] one checks that \(P^{-1}AP=B\) (since \(P^{-1}=P\)). This illustrates that even for diagonalizable operators, similarity may permute diagonal entries depending on basis choice.

7.2 Non-diagonalizable examples and Jordan blocks

Let \[ A=\begin{pmatrix}\lambda & 1\\0 & \lambda\end{pmatrix}. \] This matrix has a single eigenvalue \(\lambda\) with only one linearly independent eigenvector, so it is not diagonalizable. Any other \(2\times 2\) matrix similar to \(A\) has the same minimal polynomial \((t-\lambda)^2\) and the same Jordan block structure (one Jordan block of size \(2\)). In this sense, the existence of the superdiagonal \(1\) is not an artifact of representation: it encodes the failure of diagonalization up to similarity.

7.3 Using invariants to disprove similarity

Suppose one wants to test whether two \(2\times 2\) matrices \(A\) and \(B\) are similar. A typical approach:

compute their characteristic polynomials,
if those differ, similarity is impossible;
if characteristic polynomials match, compute minimal polynomials (or compare ranks of \(A-\lambda I\) and \(A^2-\lambda^2 I\) appropriately).

For example, two matrices might share the same characteristic polynomial but differ in minimal polynomial, indicating different Jordan structure and thus not similar.

7.4 Explicit construction of \(P\) from eigenvectors (when possible)

For diagonalizable operators, one can build \(P\) using eigenvectors. If \(A\) has eigenbasis \(\{v_1,\dots,v_n\}\) and \(B\) has eigenbasis \(\{w_1,\dots,w_n\}\) such that both represent the same eigenvalues with matching eigenvectors under a correspondence, define \(P\) by \(P v_i = w_i\). With this choice, \(AP=PB\), yielding \(B=P^{-1}AP\). When operators are not diagonalizable, one instead uses generalized eigenvectors and chains, matching Jordan blocks.

8.1 Congruence vs similarity (difference and examples)

Congruence relates bilinear or quadratic form data: \(A\) and \(B\) are congruent if \(B = P^{T} A P\) (over \(\mathbb{R}\)) or \(B=P^{*}AP\) (over \(\mathbb{C}\)), involving a transpose or adjoint rather than an inverse conjugation. Congruence preserves different invariants (e.g., signatures for real symmetric forms) and is not equivalent to similarity. Even when two matrices are similar, they need not be congruent, and vice versa.

8.2 Equivalence of bilinear forms vs similarity of operators

A linear operator can be associated to a bilinear form through matrix representations in chosen bases. Similarity captures coordinate changes for operators themselves, whereas classification of bilinear forms relies on congruence transformations. Confusing these frameworks can lead to incorrect conclusions about invariants like rank or canonical decomposition.

8.3 Similarity in abstract algebra: conjugation in groups (analogy)

In group theory, conjugation \(g \mapsto h^{-1}gh\) defines an equivalence relation, and conjugacy classes classify elements up to “change of viewpoint.” Similarity of operators is the linear-algebra counterpart: conjugation by invertible linear maps. The analogy is conceptual, emphasizing that the classification is up to an allowed reversible transformation.

8.4 Functional calculus under similarity (overview)

When an operator \(A\) is transformed by similarity \(B=P^{-1}AP\), polynomial expressions behave compatibly: \[ f(B)=P^{-1}f(A)P \] for polynomials \(f\). This means many spectral constructions computed from \(A\) carry over to \(B\) under conjugation. Under additional analytic conditions, similar statements extend to broader functional calculi (e.g., holomorphic functional calculus), but the central principle remains: similarity transports the operator while preserving the algebraic structure encoded by \(f\).