User:Prof McCarthy/Matrix similarity

For other uses, see Similarity (geometry), Similarity transformation, and Similarity (disambiguation).

Not to be confused with similarity matrix.

In linear algebra, two n-by-n matrices A and B are called similar if

$B=P^{-1}AP$

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 $A\mapsto P^{-1}AP$ 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.

Change of coordinates

The concept of matrix similarity can also be seen in the change of basis of a linear transformation.

Consider the n×n matrix A that transforms vectors x into vectors X in the n-dimensional vector space V=Rⁿ, that is

$\mathbf {X} =A\mathbf {x} ,$

where both vectors x and X are measured relative to the standard coordinate basis e₁, e₂, ..., e_n.

Now let B be an n×n matrix that changes the coordinates so the vectors x and X so they are measured relative to a different set of basis vectors, b₁, b₂, ..., b_n. Let the matrix B be constructed with b_i as its i^th column. This matrix is non-singular because its columns form a basis and are therefore linearly independent, which means the inverse of B exists. The components of x and X measured relative to the basis b_i, are given by

$\mathbf {y} =B^{-1}\mathbf {x} ,\quad {\mbox{and}}\quad \mathbf {Y} =B^{-1}\mathbf {X} .$

The transformation A can be defined in the new coordinates by the calculation,

$\mathbf {Y} =B^{-1}\mathbf {X} =B^{-1}A\mathbf {x} =B^{-1}AB\mathbf {y} ,\quad {\mbox{or}}\quad \mathbf {Y} =K\mathbf {y} ,$

where

$K=B^{-1}AB$

Thus, the matrix K defines the transformation A in the new coordinate basis, and results from a similarity transformation of A.

Eigenvector basis

In this section, it is shown that a matrix A with a linearly independent system of eigenvectors is similar to a diagonal matrix formed from its eigenvalues.

Let A be an n×n linear transformation that has n linearly independent eigenvectors v_i, and consider the change of coordinates of A so that it is defined relative to its eigenvector basis.

Recall that the eigenvectors v_i of A satisfy the eigenvalue equation,

$A\mathbf {v} _{i}=\lambda _{i}\mathbf {v} _{i}\quad i=1\ldots ,n.$

Assemble these eigenvectors into the matrix V, which is invertible because these vectors are assumed to be linearly independent. This means the coordinates x and X relative to the basis v_i can be computed as,

$\mathbf {y} =V^{-1}\mathbf {x} ,\quad {\mbox{and}}\quad \mathbf {Y} =V^{-1}\mathbf {X} .$

This yields the change of coordinates

$K=V^{-1}AV.$

To see the effect of this change of coordinates on A, introduce I=VV^-1 into the eigenvalue equation

$AV(V^{-1}\mathbf {v} _{i})=\lambda _{i}V(V^{-1}\mathbf {v} _{i}),$

and multiply both side by V^-1 to obtain

$V^{-1}AV(V^{-1}\mathbf {v} _{i})=\lambda _{i}(V^{-1}\mathbf {v} _{i}),$

Notice that

$V^{-1}\mathbf {v} _{i}=\mathbf {e} _{i},$

which is the natural basis vector. Thus,

$V^{-1}AV\mathbf {e} _{i}=K\mathbf {e} _{i}=\lambda _{i}\mathbf {e} _{i},$

and the matrix K is found to be a diagonal matrix with the eigenvalues λ_i as its diagonal elements.

This shows that a matrix A with a linearly independent system of eigenvectors is similar to a diagonal matrix formed from its eigenvalues.

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 (but not the eigenspaces, which are transformed according to the base change matrix P used).
• 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 (or any algebraically closed field), every matrix is similar to a matrix in Jordan form. Neither of these forms is unique (diagonal entries or Jordan blocks may be permuted) so they are not really normal forms; moreover their determination depends on being able to factor the minimal or characteristic polynomial of A (equivalently to find its eigenvalues). 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 (with polynomial entries) XI_n − A (the same one whose determinant defines the characteristic polynomial). Note that this Smith normal form is not a normal form of A itself; moreover it is not similar to XI_n − A either, but obtained from the latter by left and right multiplications by different invertible matrices (with polynomial entries).

Notes

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

References

• Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (Similarity is discussed many places, starting at page 44.)

Category:Matrices