1 Similarity Transformation Fundamentals

1.1 Definition via change of basis

A similarity transformation expresses the same linear map using different coordinate systems. Consider a linear transformation \(T: V \to V\) and choose two bases of the vector space, one yielding a matrix \(A\) for \(T\) and the other yielding a matrix \(B\). The matrices encode identical action of \(T\), but their entries depend on how vectors are represented in the chosen bases.

1.2 Algebraic form \(B=P^{-1}AP\)

When \(V\) is finite-dimensional and \(A\) and \(B\) are square matrices representing the same linear transformation in two bases, they satisfy \[ B = P^{-1}AP, \] where \(P\) is the change-of-basis matrix from the first basis to the second. The relation captures how coordinate vectors transform when switching from one basis to another.

1.3 Conditions for validity (invertibility of \(P\))

The formula requires \(P\) to be invertible. If the change-of-basis matrix is not invertible, it does not represent a legitimate basis change; it would correspond to a mapping that collapses dimensions and fails to provide a one-to-one correspondence between coordinate representations. In practice, similarity transformations are defined only for invertible \(P\).

1.4 Relationship to conjugation in matrix algebra

In matrix algebra, the map \(A \mapsto P^{-1}AP\) is often called conjugation by \(P\). Conjugation emphasizes that the structure of \(A\) is preserved up to “relabeling” by an invertible matrix \(P\). This perspective is useful because many invariants arise from how conjugation interacts with determinants, traces, and polynomial expressions in \(A\).

2 Basic Properties and Invariants

2.1 Preservation of eigenvalues

If \(B=P^{-1}AP\), then \(A\) and \(B\) share the same eigenvalues (including algebraic multiplicities). Intuitively, eigenvectors are carried to eigenvectors under the basis change: if \(Av=\lambda v\), then \(B(P^{-1}v)=\lambda(P^{-1}v)\). Thus, spectral data does not depend on the chosen basis.

2.2 Preservation of characteristic polynomial

The characteristic polynomial is unchanged under similarity: \[ \chi_B(t)=\chi_A(t). \] This follows because \(\det(tI-B)=\det(tI-P^{-1}AP)=\det(P^{-1}(tI-A)P)=\det(tI-A)\). Consequently, not only eigenvalues but their multiplicities are invariant.

2.3 Preservation of minimal polynomial

The minimal polynomial, defined as the monic polynomial of least degree that annihilates the matrix, is also preserved under similarity. Since matrix polynomials behave well with conjugation, the same polynomial that kills \(A\) will kill \(B\) after translating by \(P\). Therefore, the sizes and arrangement of invariant Jordan structures remain the same in a similarity class.

2.4 Preservation of rank and nullity

Similarity preserves rank and nullity because \[ B = P^{-1}AP \quad \Rightarrow \quad \ker(B) = P^{-1}\ker(A), \] and similarly for images. Rank and nullity are therefore invariants of the similarity class. More generally, ranks of powers \(A^k\) are also preserved, which ties directly to deeper invariants such as Jordan block sizes.

2.5 Trace and determinant under similarity

Both trace and determinant remain invariant: \[ \operatorname{tr}(B)=\operatorname{tr}(A), \qquad \det(B)=\det(A). \] These invariants follow from the cyclic properties of trace and the multiplicativity of determinant together with \(\det(P^{-1})\det(P)=1\). They provide quick consistency checks for purported similarity, though they are not sufficient alone.

3 Equivalence Relations and Structure

3.1 Similarity as an equivalence relation

Similarity defines an equivalence relation on square matrices. It is reflexive because choosing \(P=I\) gives \(A=I^{-1}AI\). It is symmetric because \(B=P^{-1}AP\) implies \(A=PBP^{-1}\). It is transitive because if \(B=P^{-1}AP\) and \(C=Q^{-1}BQ\), then \(C=(QP)^{-1}A(QP)\).

3.2 Transitivity and composition of transformations

The composition rule reflects how successive basis changes combine. If one changes from basis \(1\) to basis \(2\) using \(P\), and then from basis \(2\) to basis \(3\) using \(Q\), the net change from basis \(1\) to basis \(3\) is given by \(QP\). This compatibility is a key reason similarity fits naturally with the geometry of coordinate changes.

3.3 Connection to matrix conjugacy

The conjugacy viewpoint treats similarity classes as orbits under the action of the general linear group. Matrices related by conjugation represent the same operator up to a change of coordinates. This connects similarity to group actions and orbit structure, which is central in many classification problems.

3.4 When two matrices are similar: overview criteria

To decide whether two matrices are similar, one uses invariants that must match. Equality of characteristic polynomials and minimal polynomials is necessary, but not always sufficient. Further refined criteria involve matching the structure of invariant subspaces, equivalently the pattern of Jordan blocks over an algebraic closure. Over fields where canonical forms exist, similarity can be characterized by matching invariant factors in rational canonical form.

4 Worked Examples

4.1 Similarity with simple \(2\times2\) matrices

Let \[ A=\begin{pmatrix} \lambda & 0 \\ 0 & \mu \end{pmatrix}, \qquad P=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}. \] Then \[ B=P^{-1}AP \] produces a matrix that may not be diagonal, yet has the same eigenvalues \(\lambda,\mu\). This example highlights that entry-wise appearance is basis-dependent, while the underlying linear action is preserved.

4.2 Constructing \(P\) from a basis change

Suppose a linear transformation \(T\) has a matrix \(A\) in basis \(E=(e_1,\dots,e_n)\). If \(F=(f_1,\dots,f_n)\) is another basis with each \(f_i\) expressed in terms of \(E\), then \(P\) can be formed by taking as columns the coordinate vectors of \(f_i\) relative to \(E\). The similarity relation \(B=P^{-1}AP\) then follows directly from how coordinate vectors change under the basis switch.

4.3 Checking similarity computationally (conceptual steps)

A conceptual workflow is:

  1. Compute and compare invariants (characteristic and minimal polynomials).
  2. Over a suitable field, determine the Jordan structure or invariant factors.
  3. Attempt to construct \(P\) by matching generalized eigenspaces and selecting vectors that align the basis with the corresponding canonical decomposition.
  4. Verify by direct computation of \(P^{-1}AP\).

This approach balances theoretical guarantees (invariants) with constructive methods (basis alignment).

4.4 Common pitfalls (non-invertible \(P\), ordering mistakes)

Typical errors include:

  • Using a non-invertible matrix \(P\), which cannot define a basis change.
  • Mixing up the direction of basis conversion, resulting in \(B=PAP^{-1}\) rather than \(B=P^{-1}AP\).
  • Forming \(P\) with rows instead of columns (or vice versa) when assembling coordinate vectors, which leads to an incorrect similarity check.

Careful attention to conventions prevents these issues.

5 Diagonalization and Similarity

5.1 Diagonalizable matrices as a special case

A matrix is diagonalizable if it is similar to a diagonal matrix. This means there exists an invertible \(P\) such that \[ P^{-1}AP = D, \] where \(D\) is diagonal. Diagonalization is valuable because it reduces polynomial and power computations to operations on eigenvalues.

5.2 Using eigenvectors to build \(P\)

If \(A\) has a basis of eigenvectors, then one can take \(P\) to be the matrix whose columns are those eigenvectors. The similarity transformation then places \(A\) into diagonal form because the new basis aligns with the eigendirections where \(A\) acts by scaling.

5.3 Implications for powers and functions of matrices

When \(A\) is diagonalizable, \[ A^k = P D^k P^{-1}, \] and more generally any polynomial \(p(A)\) satisfies \(p(A)=P\,p(D)\,P^{-1}\). Since \(p(D)\) is obtained by applying \(p\) to each diagonal entry, computations become straightforward. This principle extends to many analytic functions defined through power series.

5.4 When diagonalization fails (geometric vs algebraic multiplicity)

Diagonalization can fail when there are not enough independent eigenvectors. The mismatch is captured by comparing:

  • algebraic multiplicity: multiplicity of \(\lambda\) as a root of the characteristic polynomial;
  • geometric multiplicity: dimension of the eigenspace \(\ker(A-\lambda I)\).

If for some eigenvalue \(\lambda\), geometric multiplicity is strictly less than algebraic multiplicity, the matrix is not diagonalizable.

6 Jordan Canonical Form

6.1 Jordan blocks and their meaning

Over an algebraically closed field, every square matrix is similar to a block diagonal matrix whose blocks are Jordan blocks. A Jordan block associated with an eigenvalue \(\lambda\) has \(\lambda\) on the diagonal, ones on the superdiagonal, and zeros elsewhere. These blocks encode both the eigenvalues and the way generalized eigenvectors contribute to the structure.

6.2 Existence and uniqueness (up to block ordering)

Jordan canonical form exists under standard hypotheses (e.g., over algebraic closure). The sizes and counts of Jordan blocks for each eigenvalue are uniquely determined by the similarity class; only the order of blocks along the diagonal may vary. Hence, Jordan form provides a complete classification up to reordering.

6.3 Similarity to Jordan form as a classification tool

Two matrices are similar over an algebraically closed field exactly when they have the same Jordan block structure (same eigenvalues with the same block sizes). This makes Jordan form a practical means of classification and theoretical means of connecting invariants (like minimal polynomials) to concrete canonical structures.

6.4 Computing Jordan form: high-level workflow

A high-level computation strategy is:

  1. Find eigenvalues and their algebraic multiplicities.
  2. Determine sizes of Jordan blocks via ranks of \((A-\lambda I)^k\) or via the dimensions of generalized eigenspaces.
  3. Build chains of generalized eigenvectors for each eigenvalue.
  4. Assemble the basis from these chains to obtain the Jordan basis, and thus the Jordan matrix.

Although implementation details vary, the workflow reflects how Jordan data is extracted from polynomial behavior of \(A\).

7 Canonical Forms and Classification by Similarity

7.1 Rational canonical form (overview)

Rational canonical form provides a similarity classification over an arbitrary field, not necessarily algebraically closed. It represents a matrix in terms of companion matrices corresponding to invariant factors, producing a block structure determined purely by algebraic information valid over the base field.

7.2 Minimal polynomial and invariant factors

The minimal polynomial corresponds to the last invariant factor in the rational canonical decomposition. The full set of invariant factors determines the similarity class: it refines information provided by eigenvalues and their multiplicities into how factors combine under the action of \(A\).

7.3 Geometric interpretation through invariant subspaces

Although rational canonical form is often presented algebraically, it can be linked to invariant subspaces. The decomposition reflects how \(V\) can be organized into subspaces stable under \(A\), with each block capturing the behavior of \(A\) restricted to a cyclic subspace.

7.4 Relation between invariants and classification

Similarity classes correspond to matching invariant data. Characteristic polynomials offer a coarse view; minimal polynomials sharpen it. Rational canonical form provides the finest invariant factor information over the chosen field, enabling complete classification without requiring eigenvalues to exist within that field.

8 Similarity and Matrix Functions

8.1 Polynomials in a matrix under similarity

For any polynomial \(p\), \[ p(B)=p(P^{-1}AP)=P^{-1}p(A)P. \] This holds because polynomial expressions are built from sums and products of \(A\), and conjugation distributes over these operations. As a result, applying a polynomial to a matrix respects similarity.

8.2 Similarity and analytic functions (overview)

Many matrix functions defined via power series inherit the same behavior: if a function \(f\) is defined in a way compatible with polynomials (e.g., through limits of polynomials), then \[ f(B)=P^{-1}f(A)P \] typically follows. Thus, the action of \(f(A)\) is basis-independent up to conjugation.

8.3 Computing \(f(A)\) via \(P^{-1}f(\text{canonical})P\)

If \(A\) is similar to a canonical form \(C\) (diagonal or Jordan), then \[ f(A)=P f(C) P^{-1}. \] For diagonal \(C\), computing \(f(C)\) reduces to evaluating \(f\) at eigenvalues. For Jordan blocks, \(f(C)\) involves derivatives and generalized eigenstructure, but still remains manageable because the canonical form has simple block structure.

8.4 Applications to matrix exponentials and dynamics (algebraic view)

Matrix exponentials \(\exp(At)\) play a central role in linear dynamical systems. Similarity implies \[ \exp(Bt)=P^{-1}\exp(At)P, \] so qualitative behaviors tied to eigenvalues (growth rates, oscillations) remain invariant under change of coordinates. Algebraically, this connects the spectrum of \(A\) to the evolution of trajectories described in different bases.

9 Connections to Linear Operators

9.1 Translating matrix similarity to operator similarity

Similarity of matrices corresponds to similarity of linear operators. Two operators are similar if there is an invertible linear map intertwining them. Under an appropriate choice of bases, operator similarity becomes exactly the matrix relation \(B=P^{-1}AP\).

9.2 Invariant subspaces and their role

Invariant subspaces provide a basis-free way to understand canonical forms. If \(W \subseteq V\) satisfies \(T(W)\subseteq W\), then restricting \(T\) to \(W\) yields a smaller matrix that forms part of the overall structure. Decompositions built from invariant subspaces lead to block forms related to similarity.

9.3 Representation independence (basis-free perspective)

The key point of similarity is that linear structure does not depend on representation. While matrices change under a basis change, the underlying operator and its invariants (spectra, polynomial identities, decomposition patterns) remain the same. This perspective supports classification and interpretation across contexts.

9.4 Modules viewpoint (brief algebraic framing)

From an algebraic viewpoint, one can treat a vector space with linear operator as a module over the polynomial ring (or related rings) where the indeterminate acts by the operator. Similarity then corresponds to module isomorphism, and canonical forms correspond to module decompositions determined by invariant factors.

10 Applications and Intuition (Algebraic)

10.1 Simplifying systems by choosing a convenient basis

Similarity is a practical tool for simplification: by moving to a basis where a matrix is diagonal, Jordan, or in rational canonical form, computations become easier. Solving linear systems, analyzing stability, and evaluating polynomial expressions all benefit from selecting a basis that exposes structure.

10.2 Understanding structure through invariants

Invariants provide a way to reason without committing to a particular basis. When two matrices share the same characteristic and minimal polynomials (and further refined canonical data), they share essential structural features. This supports both proofs and algorithm design.

10.3 Comparing linear transformations via canonical forms

Canonical forms translate the abstract notion of “same action up to basis” into a concrete format. Instead of comparing arbitrary matrices directly, one compares their canonical representatives, which are built to be easy to distinguish and classify within a similarity class.

Many decomposition results in linear algebra can be viewed through the lens of similarity. When an operator admits a decomposition into invariant components, its matrix becomes block structured after a suitable basis change. Similarity then serves as the bridge between the operator’s theoretical decomposition and the practical matrix representation.