Matrix Similarity

Matrix similarity is a relation in linear algebra between two square matrices that represent the same linear operator in possibly different coordinate systems. Matrices $A,B\in F^{n\times n}$, over a field $F$, are similar if an invertible matrix $P\in F^{n\times n}$ exists such that

The operation is called a similarity transformation. It changes the matrix representation, rather than the underlying operator. (pressbooks.cuny.edu)

Definition and change of basis

Let $T:V\to V$ act on a finite-dimensional vector space, and let $\mathcal E$ and $\mathcal F$ be ordered bases. Suppose $A$ represents $T$ in $\mathcal E$, and the columns of $P$ are the coordinates of the $\mathcal F$-basis vectors in $\mathcal E$. Then

Applying $T$ and converting its output back gives

Thus $B=P^{-1}AP$ represents $T$ in $\mathcal F$. Conversely, any invertible $P$ defines such a basis change. (pressbooks.cuny.edu)

Similarity is an equivalence relation: it is reflexive, symmetric, and transitive. Reflexivity uses the identity matrix; symmetry uses the inverse of the transforming matrix. If $B=P^{-1}AP$ and $C=Q^{-1}BQ$, then

Consequently, square matrices of a fixed size over a fixed field divide into similarity classes. (pressbooks.cuny.edu)

Preserved properties

Similar matrices have the same characteristic polynomial, because

and therefore

They consequently have the same eigenvalues, including algebraic multiplicities, as well as the same determinant and trace. Their rank is also identical, since multiplication by invertible matrices preserves rank. (ocw.mit.edu)

More generally, for every polynomial $p$,

It follows that $p(A)=0$ exactly when $p(B)=0$, so similar matrices have the same minimal polynomial: the monic polynomial of least degree annihilating the matrix. (math.uci.edu)

Eigenvectors themselves need not have identical coordinates. From $Av=\lambda v$, one obtains

The corresponding eigenspaces therefore have equal dimensions. The same calculation applies to $(A-\lambda I)^k$, preserving the dimensions of its null spaces and hence the generalized-eigenvector structure associated with each eigenvalue. (ocw.mit.edu)

Examples and limits of spectral tests

For a simple example, take

Direct multiplication gives

The matrices look different, but the calculation exhibits their similarity explicitly. This illustrates why off-diagonal entries are not similarity invariants. (sites.wcsu.edu)

Conversely, identical eigenvalues do not establish similarity. Consider

Both have characteristic polynomial $(t-1)^2$, determinant $1$, trace $2$, and rank $2$. Nevertheless, they are not similar: $P^{-1}I_2P=I_2$ for every invertible $P$. Their minimal polynomials also differ, being $t-1$ and $(t-1)^2$, respectively. Thus these commonly used invariants are necessary tests, not a complete classification. (ocw.mit.edu)

Diagonalization and canonical forms

Diagonalization is a special case of similarity. A matrix is diagonalizable over $F$ precisely when it has a basis of eigenvectors with coordinates in $F$. If those vectors form the columns of $P$, then $P^{-1}AP$ is diagonal. Two diagonalizable matrices over the same field are similar exactly when their eigenvalues agree with multiplicities. (ocw.mit.edu)

When the characteristic polynomial splits into linear factors—for example, over the complex numbers—every square matrix is similar to a Jordan normal form. Its blocks record eigenvalues and generalized-eigenvector chains. Two such matrices are similar exactly when their Jordan blocks agree, apart from block order. Diagonalizability corresponds to all blocks having size one. (pmelvin.blogs.brynmawr.edu)

Over an arbitrary field, rational canonical form provides a complete classification without requiring the eigenvalues to belong to that field. Its invariant-factor polynomials determine a unique canonical representative, and two matrices are similar exactly when these representatives coincide. (pmelvin.blogs.brynmawr.edu)

Testing similarity and numerical computation

The defining equation can be rewritten as

For fixed $A$ and $B$, this is a homogeneous system of linear equations in the entries of $P$. Similarity holds precisely when its solution space contains an invertible matrix. A nonzero solution alone is insufficient, since it may be singular. Canonical forms offer an alternative exact test. (sites.wcsu.edu)

In numerical linear algebra, Schur decomposition uses restricted similarity transformations. A complex matrix has a representation $A=QTQ^*$, with $Q$ a unitary matrix and $T$ upper triangular; $Q^*$ denotes the conjugate transpose. For real matrices, $Q$ can be an orthogonal matrix, while $T$ is block upper triangular with diagonal blocks of size one or two. These forms support practical eigenvalue computation while retaining the original eigenvalues. (netlib.org)