Similarity is an equivalence relation on the space of square matrices.
Similar matrices share many properties:
If u is an eigenvector of A, then P−1u is an eigenvector of B. We need to prove that .
- Characteristic polynomial
- Minimal polynomial (among the other similarity invariants in the Smith normal form)
- Elementary divisors
There are two reasons for these facts:
- Two similar matrices can be thought of as describing the same linear map, but with respect to different bases
- The map X ↦ P−1XP is an automorphism of the associative algebra of all n-by-n matrices, as the one-object case of the above category of all matrices.
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. Another normal form, the rational canonical form, works over any field. By looking at the Jordan forms or rational canonical forms of A and B, one can immediately decide whether A and B are similar. The Smith normal form can be used to determine whether matrices are similar, though unlike the Jordan and rational canonical forms, a matrix is not necessarily similar to its Smith normal form.
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 quite useful: one may safely enlarge the field K, for instance to get an algebraically closed field; Jordan forms can then be computed over the large field and can be used to determine whether the given matrices are similar over the small field. This approach can be used, for example, to show that every matrix is similar to its transpose.
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.
In group theory similarity is called conjugacy. In category theory, given any family Pn of invertible n-by-n matrices defining a similarity transformation for all rectangular matrices sending the m-by-n matrix A into Pm−1APn, the family defines a functor that is an automorphism of the category of all matrices, having as objects the natural numbers and morphisms from n to m the m-by-n matrices composed via matrix multiplication.
- Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (Similarity is discussed many places, starting at page 44.)