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Several equivalence relations in mathematics are called similarity. For similarity between people, see similarity (psychology). In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
In social psychology, similarity refers to how closely attitudes, values, interests and personality match between people. ...
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. In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...
In Euclidean geometry, scaling is an affine, linear transformation that can enlarge or diminish an object by certain factors. ...
In geometry, two objects are of the same shape if one can be transformed to another (ignoring color) by dilating (that is, by multiplying all distances by the same factor) and then, if necessary, rotating and translating. ...
Mirror Image is an episode of the television series The Twilight Zone. ...
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. In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ...
In plane geometry, a square is a polygon with four equal sides and equal angles. ...
A parabola The parabola (from the Greek: παÏαβολή) is a conic section generated by the intersection of a cone and a plane tangent to the cone or parallel to some plane tangent to the cone. ...
In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ...
A graph of a hyperbola, where h = k = 0 and a = b = 2. ...
For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
This article is about angles in geometry. ...
Formally, we define a similarity or similarity transformation (also called dilation) 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 In mathematics, a dilation is a function f from a metric space into itself that satisfies the identity where d(x,y) is the distance from x to y and r is some real number. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
Scalar is a concept that has meaning in mathematics, physics, and computing. ...
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. In mathematics the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...
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. In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin. ...
In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...
The 2D similarity transformations expressed in terms of the complex plane are f(z) = az + b and , and all affine transformations are of the form (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.) In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
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. A self-similar object is exactly or approximately similar to a part of itself. ...
Value A logarithmic scale is a scale of measurement that gives the logarithm of a physical quantity instead of the quantity itself. ...
Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...
Similar triangles If triangle ABC is similar to triangle DEF, then this relation can be denoted as - .
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: There are several meanings for the word deduction: Natural deduction Deductive reasoning Deductions in terms of taxation, such as Itemized deductions Standard deduction See also: Logic Venn diagram Inductive reasoning Both statistics and the scientific method rely on both induction and deduction. ...
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. See: Equality (mathematics) Social equality Equality, Illinois This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
A graph with 6 vertices (nodes) and 7 edges. ...
- 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.
See also Congruence (geometry). In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...
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 Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
For the square matrix section, see square matrix. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
- P −1AP = B.
A similarity transformation is such a transformation of a matrix A into a matrix 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: In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...
In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...
The minimal polynomial of an n-by-n matrix A over a field F is the polynomial p(x) with leading coefficient 1 over F of least degree such that p(A)=0. ...
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. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ...
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...
In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ...
In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K, to what extent M can be simplified into a standard...
In linear algebra, the Frobenius normal form of a matrix is a normal form that reflects the structure of the minimal polynomial of a matrix. ...
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 abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...
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. In linear algebra, a permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. ...
In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
A complex square matrix A is a normal matrix iff where A* is the conjugate transpose of A (if A is a real matrix, this is the same as the transpose of A). ...
Another important equivalence relation for real matrices is congruency. In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...
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). Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
personal space, proxemics. ...
The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are
- Positive defined:
- Majored by the similarity of one element on itself (auto-similarity): and
More properties can be invoked, such as reflectivity () or finiteness (). The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).
See also This article is about proportionality, the mathematical relation. ...
In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...
Semantic similarity, variously also called semantic closeness/proximity/nearness, is a concept whereby a set of documents or terms within term lists are assigned a metric based on the likeness of their meaning / semantic content. ...
External links - Three Similar Triangles
- Directly Similar Figures
- Napoleon's Propeller
- Two Triples of Similar Triangles
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