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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 scalarAbstract algebra Algebra Linear algebra The concept of a scalar is used in mathematics and physics. The concept used in physics is a more concrete version of the same idea that goes by that name in mathematics. In mathematics, the meaning of scalar depend r, so that for any two points x and y we have
where "d(x,y)" is the Euclidean distanceThe Euclidean distance of two points x x . x and y y . y in Euclidean n space is computed as : It 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 from x to y. Two sets are called similar if one is the image of the other under such a similarity.
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 deduceSee natural deduction Deductive reasoning See also: logic Venn diagram inductive reasoning Both statistics and the scientific method rely on both induction and deduction. proportionalities between corresponding sides of the two triangles, such as the following:
In linear algebraLinear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is wi, two n-by-n matricesAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number A and B over the fieldIn 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 famil K are called similar if there exists an invertibleIn 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 I where I denotes the n by n identity matrix and the multiplication used is ordinary matr n-by-n matrix P over K such that
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:
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.