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Howell Normal Form
In linear algebra and ring theory, the Howell normal form is a generalization of the row echelon form of a matrix over \Z_N, the ring of integers modulo N. The row spans of two matrices agree if, and only if, their Howell normal forms agree. The Howell normal form generalizes the Hermite normal form, which is defined for matrices over \Z. Definition A matrix A \in \Z_N^ over \Z_N is called to be in ''row echelon form'' if it has the following properties: * Let r be the number of non-zero rows of A. Then the topmost r rows of the matrix are non-zero, * For 1 \leq i \leq r, let j_i be the index of the leftmost non-zero element in the row i. Then j_1 < j_2 < \dots < j_r. With elementary transforms, each matrix in the row echelon form can be reduced in a way that the following properties will hold: * For each , the leading element i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Row Echelon Form
In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore, only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices. Specifically, a matrix is in row echelon form if * All rows consisting of only zeroes are at the bottom. * The leading entry (that is the left-most nonzero entry) of every nonzero row is to the right the leading entry of every row above. Some texts add the condition that the leading coefficient must be 1 while others regard this as ''reduced'' row echelon form. These two conditions imply that all entries in a column below a leadin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Integers Modulo N
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Normal Form
In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R''n'', the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only. Other applications of the Hermite normal form include integer programming, cryptography, and abstract algebra. Definition Various authors may prefer to talk about Hermite normal form in either row-style or column-style. They are essentially the same up to transposition. Row-style Hermite normal form An m by n matrix A with integer entries has a (row) Hermite normal form H if there is a square unimodular matrix U where H=UA and H has the following restrictions: # H is upper triangular (that is, h''ij'' = 0 for ''i'' > ''j''), and any rows of zeros are located below any other row. # The leading c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Matrix
In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss–Jordan elimination to further reduce the matrix to reduced row echelon form. Elementary row operations There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations): ;Row switching: A row within the matrix can be switched with another row. : R_i \leftrightarrow R_j ;Row multiplication: Each element in a row can be multiplied by a non-zero constant. It is also known as ''scaling'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized either as the intersection of all linear subspaces that contain , or as the smallest subspace containing . The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules. To express that a vector space is a linear span of a subset , one commonly uses the following phrases—either: spans , is a spanning set of , is spanned/generated by , or is a generator or generator set of . Definition Given a vector space over a field , the span of a set of vectors (not necessarily infinite) is defined to be the intersection of all subspaces of that contain . is referred to as the subspace ''spanned by'' , or by the vectors in . Conversely, is called a ''spanning set'' of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often denoted by O_K or \mathcal O_K. Since any integer belongs to K and is an integral element of K, the ring \mathbb is always a subring of O_K. The ring of integers \mathbb is the simplest possible ring of integers. Namely, \mathbb=O_ where \mathbb is the field of rational numbers. And indeed, in algebraic number theory the elements of \mathbb are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers \mathbb /math>, consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field \mathbb(i) of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
