In linear algebra, two n-by-n matrices A and B are called similar if
for some invertible n-by-n matrix P. Similar matrices represent the same
linear operator under two different bases, with P being the change of basis
matrix. A transformation is called a similarity
transformation or conjugation of the matrix A. In the general linear group,
similarity is therefore the same as conjugacy, and similar matrices are also
called conjugate; however in a given subgroup H of the general linear group,
the notion of conjugacy may be more restrictive than similarity, since it
requires that P can be chosen to lie in H.
Properties Similarity is an equivalence relation on
the space of square matrices. Similar matrices share any properties
that are really properties of the represented linear operator:
Rank Characteristic polynomial, and
attributes that can be derived from it: Determinant
Trace Eigenvalues, and their algebraic
multiplicities Geometric multiplicities of eigenvalues.
Minimal polynomial Elementary divisors, which form a
complete set of invariants for similarity
Rational canonical form Because of this, for a given matrix A,
one is interested in finding a simple "normal form" B which is similar to
A—the study of A then reduces to the study of the simpler matrix B. For
example, A is called diagonalizable if it is similar to a diagonal matrix. Not
all matrices are diagonalizable, but at least over the complex numbers, every
matrix is similar to a matrix in Jordan form. Neither of these forms is unique
so they are not really normal forms; moreover their determination depends on
being able to factor the minimal or characteristic polynomial of A. The
rational canonical form does not have these drawbacks: it exists over any
field, is truly unique, and it can be computed using only arithmetic
operations in the field; A and B are similar if and only if they have the
same rational canonical form. The rational canonical form is determined by
the elementary divisors of A; these can be immediately read off from a matrix in
Jordan form, but they can also be determined directly for any matrix by
computing the Smith normal form, over the ring of polynomials, of the matrix
XIn − A. Note that this Smith normal form is not a normal form of A itself;
moreover it is not similar to XIn − A either, but obtained from the latter by
left and right multiplications by different invertible matrices.
Similarity of matrices does not depend on the base field: if L is a field
containing K as a subfield, and A and B are two matrices over K, then A and B
are similar as matrices over K if and only if they are similar as matrices
over L. This is so because the rational canonical form over K is also the
rational canonical form over L. This means that one may use Jordan forms that
only exist over a larger field to determine whether the given matrices are
similar. In the definition of similarity, if the
matrix P can be chosen to be a permutation matrix then A and B are
permutation-similar; if P can be chosen to be a unitary matrix then A and B are
unitarily equivalent. The spectral theorem says that every normal matrix is
unitarily equivalent to some diagonal matrix. Specht's theorem states that two
matrices are unitarily equivalent if and only if they satisfy certain trace
equalities. See also
Matrix congruence Matrix equivalence
Canonical forms Notes
References Beauregard, Raymond A.; Fraleigh, John
B., A First Course In Linear Algebra: with Optional Introduction to Groups,
Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X
Bronson, Richard, Matrix Methods: An Introduction, New York: Academic Press,
LCCN 70097490 Horn and Johnson, Matrix Analysis,
Cambridge University Press, 1985. ISBN 0-521-38632-2.