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Integer Matrix
In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics. Examples :\left(\begin 5 & 2 & 6 & 0\\ 4 & 7 & 3 & 8\\ 5 & 9 & 0 & 4\\ 3 & 1 & 0 & \!\!\!-3\\ 9 & 0 & 2 & 1\end\right) and \left(\begin 1 & 5 & 0\\ 0 & 9 & 2\\ 1 & 7 & 3\end\right) are both examples of integer matrices. Properties Invertibility of integer matrices is in general more numerically stable than that of non-integer matrices. The determinant of an integer matrix is itself an integer, thus the numerically smallest possible magnitude of the determinant of an invertible integer matrix is one, hence where inverses exist they do not become excessively large (see condition number). Theorems from matrix theory that infer properties from determinants thu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating Point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be represented as a base-ten floating-point number: 12.345 = \underbrace_\text \times \underbrace_\text\!\!\!\!\!\!^ In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. The term ''floating point'' refers to the fact that the number's radix point can "float" anywhere to the left, right, or between the significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation. A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom. For this reason, floating-poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodular Matrix
In mathematics, a unimodular matrix ''M'' is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix ''N'' that is its inverse (these are equivalent under Cramer's rule). Thus every equation , where ''M'' and ''b'' both have integer components and ''M'' is unimodular, has an integer solution. The ''n'' × ''n'' unimodular matrices form a group called the ''n'' × ''n'' general linear group over \mathbb, which is denoted \operatorname_n(\mathbb). Examples of unimodular matrices Unimodular matrices form a subgroup of the general linear group under matrix multiplication, i.e. the following matrices are unimodular: * Identity matrix * The inverse of a unimodular matrix * The product of two unimodular matrices Other examples include: * Pascal matrices * Permutation matrices * the three transformation matrices in the ternary tree of primitive Pythagorean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GCD Matrix
In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix. Definition Let S=(x_1, x_2,\ldots, x_n) be a list of positive integers. Then the n\times n matrix (S) having the greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ... \gcd(x_i, x_j) as its ij entry is referred to as the GCD matrix on S.The LCM matrix /math> is defined analogously. The study of GCD type matrices originates from who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the n\times n matrix (\gcd(i,j)) is \phi(1)\phi(2)\cdots\phi(n), where \phi is Euler's totient function. Bourque–Ligh conjecture conjectured that the LCM matrix on a GCD-closed set S is nonsingular. This conjectu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nth Root
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'': :r^n = x, where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root of degree 2 is called a ''square root'' and a root of degree 3, a ''cube root''. Roots of higher degree are referred by using ordinal numbers, as in ''fourth root'', ''twentieth root'', etc. The computation of an th root is a root extraction. For example, 3 is a square root of 9, since 3 = 9, and −3 is also a square root of 9, since (−3) = 9. Any non-zero number considered as a complex number has different complex th roots, including the real ones (at most two). The th root of 0 is zero for all positive integers , since . In particular, if is even and is a positive real number, one of its th roots is real and positive, one is negative, and the others (when ) are non-real complex numbers; if is even and is a negative real numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel–Ruffini Theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means that the coefficients of the equation are viewed and manipulated as indeterminates. The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, (which was refined and completed in 1813 and accepted by Cauchy) and Niels Henrik Abel, who provided a proof in 1824. ''Abel–Ruffini theorem'' refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from . Galois theory implies also that :x^5-x-1=0 is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field , denoted by , is the intersection of and : it can also be characterised as the maximal order of the field . Each algebraic integer belongs to the ring of integers of some number field. A number is an algebraic integer if and only if the ring \mathbbalpha/math> is finitely generated as an abelian group, which is to say, as a \mathbb-module. Definitions The following are equivalent definitions of an algebraic integer. Let be a number field (i.e., a finite extension of \mathbb, the field of rational numbers), in other words, K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Permutation Matrices
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is :\begin 0 & 0 & 3 & 0\\ 0 & -7 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt2\end. Structure An invertible matrix ''A'' is a generalized permutation matrix if and only if it can be written as a product of an invertible diagonal matrix ''D'' and an (implicitly invertible) permutation matrix ''P'': i.e., :A = DP. Group structure The set of ''n'' × ''n'' generalized permutation matrices with entries in a field ''F'' forms a subgroup of the general linear group GL(''n'', ''F''), in which the group of nonsingular diagonal matrices Δ(''n'', ''F'') forms a normal subgroup. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by Function composition, composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrix, orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose invertible matrix, inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact group, compact. The orthogonal group in dimension has two connected component (topology), connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
