Similarity mathematics

Several equivalence relations in mathematics are called similarity. The first one discussed below is about similar objects, the second one about similar matrices. The term "similarity transformation" has two different meanings, each related to one of the meanings of "similar".

For similarity between people, see similarity (psychology).

Geometry

Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. One can be obtained from the other by uniformly "stretching", possibly with additional rotation, i.e., both have the same shape, or additionally the mirror image is taken, i.e., one has the same shape as the mirror image of 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.

One of the meanings of the terms similarity and similarity transformation (also called dilation) of a Euclidean space is 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.

A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image. A similarity is a composition of a homothety and an isometry.

The 2D similarity transformations expressed in terms of the complex plane are $f(z)=az+b$ and $f(z)=a\overline z+b$, and all affine transformations are of the form $f(z)=az+b\overline z+c$ (a, b, and c complex). (This paragraph views the complex plane as a 2-dimensional space over the reals; over the complex field, it is 1-dimensional.)

Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set {.., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, ..}. When this set is plotted on a logarithmic scale is has translational symmetry.

Similar triangles

If triangle ABC is similar to triangle DEF, then this relation can be denoted as $\triangle ABC\sim \triangle DEF$.

In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°.

Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent with the angle at F. Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following: ${AB \over BC}={DE \over EF},$ ${AB \over AC}={DE \over DF},$ ${AC \over BC}={DF \over EF},$ ${AB \over DE}={BC \over EF}={AC \over DF}.$

Angle/side similarities

A concept commonly taught in high school mathematics is that of proving the angle and side theorems, which can be used to define two triangles as similar (or indeed, congruent).

In each of these three-letter acronyms, A stands for equal angles, and S for sides. For example, ASA denominates an angle, side and angle that are all equal and adjacent, in that order.

• AAA - Angle-Angle-Angle. If two triangles share three common angles, they are similar. (Obviously, this means that the side lengths are locked in a common ratio, but can vary proportionally, making the triangles similar.) . Additionally, knowing that that the interior angles of a triangle have a sum of 180 degrees, having two triangles with only two common angles (sometimes known as AA) implies similarity as well.

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.

One of the meanings of the term "similarity transformation" is such a transformation of a matrix A into a matrix B.

In group theory similarity is called conjugacy.

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 $\mapsto$ 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.

Another important equivalence relation for real matrices is congruency.

Two real matrices A and B are called congruent if there is a regular real matrix P such that

PTAP = B.

Topology

In topology, a metric space can be constructed by defining a similarity instead of a distance. The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of disimilarity: the closer the points, the lesser the distance).

The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are

1. Positive defined: $\forall (a,b),S(a,b)\geq 0$
2. Majored by the similarity of one element on itself (auto-similarity): $S(a,b)\leq S(a,a)$ and $\forall (a,b),S(a,b)=S(a,a)\Leftrightarrow a=b$

More properties can be invoked, such as reflectivity ( $\forall (a,b)\ S(a,b)=S(b,a)$) or finiteness ( $\forall (a,b)\ S(a,b)<\infty$). The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).