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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 Circle s are similar to each other, all Square s are similar to each other, and all Parabola s are similar to each other. On the other hand, Ellipse s are ''not'' all similar to each other, ''nor'' are Hyperbola s all similar to each other. Two Triangles are similar if and only if they have the same three Angle s, the so-called "AAA" condition.


Similar triangles

If triangle ''ABC'' is similar to triangle ''DEF'', then this relation can be denoted as
: riangle ABC \sim riangle 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}.

This idea can be extended to similar Polygon s with any number of sides. That is, given any two similar polygons, the corresponding sides are proportional.


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 equal sides. For example, ASA refers to 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, since the interior angles of a triangle have a sum of 180°, having two triangles with only two common angles (sometimes known as '''AA''') implies similarity as well.


''See also:'' Congruence (geometry)


Similarity in Euclidean space


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)) = r d(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 .

Viewing the Complex Plane as a 2-dimensional space over the Reals , 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 Transformation s are of the form f(z)=az+b\overline z+c (''a'', ''b'', and ''c'' complex).


Similarity in general metric spaces


In a general Metric Space (X,d), an exact similitude is a Function ''f'' from the metric space X 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)) = r d(x,y), \,


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


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 Eigenvalue s (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''−1''XP'' 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 Number s (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 .

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

P



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 '''dissimilarity:''' 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
# Positive defined: orall (a,b), S(a,b)\geq 0
# Majored by the similarity of one element on itself (auto-similarity): S (a,b) \leq S (a,a) and orall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b

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


SELF-SIMILARITY

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 it has Translational Symmetry .


SEE ALSO



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