Similarity (mathematics)

Several equivalence relations in mathematics are called similarity.

Geometry

Two geometrical objects are called similar if both have the same shape. One can be obtained from the other by uniformly "stretching", i.e. one is congruent to an "enlargement" of the other, or the mirror image of one has the same shape as the other.

For example, all circles are similar to each other, all squares are similar to each other, and all parabolas are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. Two triangles are similar if and only if they have the same three angles, the so-called "AAA" condition.

Formally, we define a similarity or similarity transformation of a Euclidean space as a function f from the space into itself that multiplies all distances by the same positive scalar r, so that for any two points x and y we have

d(f(x),f(y)) = rd(x,y),

where "d(x,y)" is the Euclidean distance from x to y. Two sets are called similar if one is the image of the other under such a similarity.

Linear algebra

In linear algebra, two n-by-n matrices A and B over the field K are called similar if there exists an invertible n-by-n matrix P over K such that

P ⁻¹AP = B.

Similar matrices share many properties: they have the same rank, the same determinant, the same trace, the same eigenvalues (but not necessarily the same eigenvectors), the same characteristic polynomial and the same minimal polynomial. 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

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.

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.

If in the definition of similarity, 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.