Similarity (mathematics) 

Geometry

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

For example, all circles are similar, as are all squares. Two triangles are similar if and only if they have the same three angles, the so-called "AAA" condition.

Formally, we define a similarity 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

<math>d(f(x),f(y))=rd(x,y),</math>

Linear algebra

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

P ^-1AP = B.

• 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

If in the definition of similarity, the matrix P can be choses 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.

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