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Egwald Mathematics - Linear Algebra: Matrices and Matrix Decomposition Permutation, rotation, and reflection matrices are orthonormal matrices. Multiplying a column or row by a constant scalar multiplies the determinant by the constant scalar. Matrices with fewer rows than columns are found in under-determined problems, with fewer equations than unknown variables. www.egwald.com /linearalgebra/matrices.php   (3984 words)

Matrix (mathematics) - Wikipedia, the free encyclopedia Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on two parameters. Matrices can be added, multiplied, and decomposed in various ways, marking them as a key concept in linear algebra and matrix theory. Matrices over a polynomial ring are important in the study of control theory. en.wikipedia.org /wiki/Matrix_(mathematics)   (1664 words)

General linear group - Wikipedia, the free encyclopedia The Lie algebra corresponding to GL(n,C) consists of all n×n complex matrices with the commutator serving as the Lie bracket. Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix. The center of SL(n, F) is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of nth roots of unity in the field F. en.wikipedia.org /wiki/General_linear_group   (1513 words)

PlanetMath: diagonal matrix The identity matrix and zero matrix are diagonal matrices. It follows that real (and complex) diagonal matrices are normal matrices. Diagonal matrices are also sometimes called quasi-scalar matrices [1]. planetmath.org /encyclopedia/QuasiScalarMatrices.html   (200 words)

Math.com Online Solvers Matrices   (Site not responding. Last check: 2007-11-03) Matrices are added to and subtracted from one another element by element. For instance, when adding two matrices A and B, the element at row 1, column1 of A is added to the element at row 2, column 2 of B to give the element at row 1, column 1 of the answer. Multiplying a matrix by a scalar simply involves multiplying each element by that scalar, whilst raising a matrix to a positive integer power can be achieved by a series of matrix multiplications. www.math.com /students/solvers/matrices/matrices.htm   (274 words)

QuickMath Automatic Math Solutions Matrices are added to and subtracted from one another element by element. For instance, when adding two matrices A and B, the element at row i, column j of A is added to the element at row i, column j of B to give the element at row i, column j of the answer. Multiplying a matrix by a scalar simply involves multiplying each element by that scalar, whilst raising a matrix to a positive integer power can be achieved by a series of matrix multiplications. www.quickmath.com /www02/pages/modules/matrices   (274 words)

[No title]   (Site not responding. Last check: 2007-11-03) Matrices allow not only operations characteristic of linear transformations, (addition, or adding two or more matrices together, and scalar multiplication, multiplying an entire matrix by a real scalar,) but also allows, in certain cases, multiplication of vectors and finding the inverse of the linear tranformation. Multiplying matrices, solving a system of equations, determining if a given linear transformation has an inverse, finding such an inverse; all these are fundamental aspects of linear algebra that are reduced to trivial, yet tedious, calculations through matrices. Functionality - Addition of matrices Multiplying a matrix by a scalar Multiplying a matrix by another matrix Inverting a matrix, when possible Challenges - **The biggest advantage of a matrix is that it can handle large sets of data. www.people.fas.harvard.edu /~dpopper/fpc/proposal.txt   (1286 words)

SparkNotes: Matrices: Introduction and Summary Matrices will be used to organize data as well as to solve for variables. The second section explains two types of multiplication associated with matrices: scalar multiplication--that is, multiplication by a constant--and multiplication of two matrices. Matrices are also used by mathematicians, physicists, and biologists to organize data and study complex phenomena; for example, matrices are used to study population growth and determine when a population will stabilize. www.sparknotes.com /math/algebra2/matrices/summary.html   (378 words)

PH@School: UCSMP: Adv. Algebra: Teacher Chapter 4 Matrices first entered high school mathematics programs with the new-math curricula of the late 1950s and early 1960s. Lessons 4-1 to 4-3 introduce the vocabulary and notation for matrices, their use in storing data, and the operations of matrix addition and multiplication. Lessons 4-4 and 4-5 discuss matrices used for size and scale changes, along with those properties which are preserved or not preserved. www.phschool.com /atschool/ucsmp/adv_algebra/Teacher_Area/ADV_TC4.html   (373 words)

[ref] 24 Matrices Because matrices are just a special case of lists, all operations and functions for lists are applicable to matrices also (see chapter Lists). The rules for arithmetic operations involving matrices are in fact special cases of those for the arithmetic of lists, given in Section Arithmetic for Lists and the following sections, here we reiterate that definition, in the language of vectors and matrices. More general situations are for example the sum of an integer scalar and a matrix over a finite field, or the sum of a finite field element and an integer matrix. www.gap-system.org /Manuals/doc/htm/ref/CHAP024.htm   (4165 words)

John Halleck's Matrices Matrices of the same size may be added, by making a new matrix of the same size, with elements that just add the corresponding elements from the matrices being added. Unlike regular arithmetic, A*B with matrices is not the same as B*A. (The proper term for this is that "matrix multiplication does not commute") Because matrix multiplication does not commute, we have to distinguish between premultiplying by M (as in M*P) and postmultiplying by M (as in P*M). There is a lot of very special mathematics that goes with this, but the short answer is that positive definite matrices are very easy to find the inverses of, and are very easy to deal with, and show up all over the place when doing least squares methods. www.cc.utah.edu /~nahaj/math/matrices.html   (2427 words)

Matrix Tutorial 1: Stochastic Matrices This tutorial introduces matrices, eigenvalues, and eigenvectors to IR students and search engine marketers. If all elements of a scalar matrix are 1 this is termed a unit matrix or an identity matrix, I. The expression doubly stochastic matrix is reserved for square matrices whose sums of both rows and columns equal 1. www.miislita.com /information-retrieval-tutorial/matrix-tutorial-1-stochastic-matrices.html   (2152 words)

GameDev.net - Vectors and Matrices: A Primer Matrices are very powerful, and form the basis of all modern computer graphics, the advantage of them being that they are so fast. Matrices can be of any dimensions, but in terms of computer graphics, they are usually kept to 3x3 or 4x4. These two matrices are conformable because the number of rows in A is the same as the number of columns in B. www.gamedev.net /reference/articles/article1832.asp   (3700 words)

Matrices   (Site not responding. Last check: 2007-11-03) Two matrices are equal if they have the same dimensions and if the corresponding elements are equal. Corresponding elements are two or more numbers which are in the same row and the same column of their respective matrices (for example, two elements which would correspond are numbers located in the third row and fourth column of two different matrices). Matrices are very useful because they are a fast and easy was of solving systems of linear equations. josh.tumaz.com /jjs_math/Matrices/index.html   (964 words)

[ref] 22 Matrices   (Site not responding. Last check: 2007-11-03) Note that we want that Lie matrices shall be matrices that behave in the same way as ordinary matrices, except that they have a different multiplication. It is allowed that one operand is a finite field element, a finite field vector, a finite field matrix, or a list of finite field matrices, and the other operand is an integer, an integer vector, an integer matrix, or a list of integer matrices. Block matrices are a special representation of matrices which can save a lot of memory if large matrices have a block structure with lots of zero blocks. www.math.psu.edu /local_doc/gap4/htm/ref/CHAP022.htm   (2165 words)

NMath Core User's Guide - 5.5 Arithmetic Operations on Matrices NMath Core provides overloaded arithmetic operators for matrices with their conventional meanings for those.NET languages that support them, and equivalent named methods for those that do not. All binary operators and equivalent named methods work either with two matrices, or with a matrix and a scalar. Matrices must have the same dimensions to be combined using the element-wise operators. www.centerspace.net /doc/NMath/Core/user/matrix6.html   (388 words)

Linear Algebra Review One common convention is to denote matrices with boldface uppercase letters, vectors with boldface lowercase letters, and scalars with italic lowercase letters. Since vectors are just "degenerate" matrices, we can now see that in order to get the inner product of two vectors, we have to treat the first one as a row vector (1 x n) and the second one as a column vector (n x 1), the result being a 1 x 1 matrix (i.e. One way to create more general multi-dimensional rotation matrices is by combining such matrices, each of which accomplishes a rotation in a single plane defined by a pair of coordinates. www.ling.upenn.edu /courses/Spring_2003/ling525/linear_algebra_review.html   (5601 words)

u_eigen However, if we are willing to use complex numbers then these matrices are like the previous case; they do have complex eigenvalues with corresponding eigenvectors. Matrices that have only one eigenvalue, other than scalar matrices. matrices with only one eigenvalue are called ``defective''. www.math.ucla.edu /~baker/115ah.1.01f/handouts/u_eigen/node9.html   (343 words)

[No title] 1.10.2 Identity matrices: An identity matrix is a square matrix with 1s on the principal diagonal and 0s in all other elements of the array. In the case where A is a square matrix, however, I will have the same dimension in each case and is an example of an exception to the rule that matrix multiplication is not commutative. Note also that the null matrices may each be of different dimension even though they are each denoted by 0. www.lancs.ac.uk /people/ecajj/213l1.doc   (2285 words)

How to Use INDSCAL, a Computer Program for Canonical Decomposition of N-way Tables and Individual Differences in ... In this case, rectangular data matrices may be analyzed, identifying the underlying dimensions of each of the n dimensional input matrices. INDIFF treats the first of the N matrices into which the N-way table is decomposed as the subjects' weight matrix and matrices 2 and 3 as stimulus matrices. That is the program requires that the initial matrices 2 through N be either supplied by the user or generated in the program. marketing.byu.edu /htmlpages/books/pcmds/INDSCAL.html   (2005 words)

6.2 - Operations with Matrices A scalar is a number, not a matrix. Since the order (dimensions) of the matrices don't have to be the same, there may not be corresponding elements to multiply together. The constant 3 is not a matrix, and you can't add matrices and scalars together. www.richland.edu /james/lecture/m116/matrices/operations.html   (833 words)

Stat/Math - More on Matrices As we saw earlier, +, -, *, and / are defined in an intuitive manner for matrices. All operation involving a scalar and a matrix affect the matrix on an entry-by-entry basis, with one exception: the power ("^") operator. One of the main uses of matrices is in representing systems of linear equations. www.indiana.edu /~statmath/math/matlab/gettingstarted/morematrices.html   (610 words)

MATH 137   (Site not responding. Last check: 2007-11-03) If two matrices are not of the same size (dimension), they cannot be added. Subtraction is performed on matrices of the same size by subtracting corresponding elements. Scalar multiplication is the operation of multiplying a matrix by a number (scalar). www.brookdale.cc.nj.us /fac/math/eklett/Class_lecture_7.htm   (531 words)

making matrices Scalars are converted to scalar matrices when necessary. There are commands for horizontal and vertical concatenation of matrices, and again, scalars are converted to scalar matrices when necessary. The degree of a tensor product of two matrices is the sum of the degrees, and its source module is the tensor product of the source modules. www.msri.org /about/computing/docs/macaulay/2-0.9/1204.html   (505 words)

Calculating Matrices All the fundamental properties of matrices, that is, all the properties that every matrix possesses, come from the definition as an array of transformation coefficients. This is one way in which matrices may assume properties in addition to the basic ones, and is one of the most important applications of matrices. Matrices can be moved around on the stack using any of the stack commands, and they occupy only one level. www.du.edu /~jcalvert/tech/matrix48.htm   (5670 words)

Block Toeplitz Matrices These are not necessarily scalar Toeplitz matrices, but, similarly to those, the block structure can be used in a fast matrix-vector multiplication. The dimension of the blocks (n) was as close as possible to the block dimension (N) of the matrix, while still keeping both n and N a power of two. The great difference to the scalar case is due to the extra overhead needed and that the vectors used in the FFT routine now are of length 2n and not 2Nn, as they would be in the scalar case. www.mai.liu.se /~evlun/pub/lic/node11.html   (1230 words)

Some Elementary Concepts   (Site not responding. Last check: 2007-11-03) Three fundamental concepts in MATLAB, and in linear algebra, are scalars, vectors and matrices. Array is just a generic word for matrices and vectors; a vector is simply a one-dimensional array, while a matrix is a two-dimensional array --- so remember this when consulting the MATLAB help texts. You can also create matrices from row vectors or column vectors, as long as all of the vectors being used to create the matrix have the same number of elements. www.engr.iupui.edu /matlabtutorial/introductory_lessons/some_elementary_concepts.htm   (2252 words)

GNU Emacs Calc 2.02 Manual: Mode Settings In this mode, all objects are assumed to be matrices unless provably otherwise. Another way to mix scalars and matrices is to use selections (see section 12.1 Selecting Sub-Formulas). Matrices are displayed in a multi-line tabular format, but all other objects are written in linear form, as they would be typed from the keyboard. www.xemacs.org /Documentation/packages/html/calc_9.html   (9366 words)

Non-local densities The density matrices in the spin and isospin spaces can be expressed as linear combinations of the unity and Pauli matrices. Since the p-h density matrix and the Pauli matrices are both hermitian, all the p-h densities are hermitian too, Equations (28) and (34) are fulfilled independently of any other symmetries conserved by the system; they result from general properties (11) of density matrices www.fuw.edu.pl /~dobaczew/nppair60w/node4.html   (356 words)

Construction of Matrix Algebras and their Elements In either format, if q=p^e, where p is prime and e>1, then an entry x is written as a vector using the base-p representation of length e of x and the corresponding element in K is used (see the Finite Fields chapter for details). Given a commutative ring S and a positive integer n, create the S-algebra R consisting of the n x n matrices over the ring S generated by the elements defined in the list L. Let F denote the algebra M_n(S). Repetitions of an element and occurrences of scalar matrices are removed. www.umich.edu /~gpcc/scs/magma/text912.htm   (936 words)

An introduction to MATRICES A scalar matrix S is a diagonal matrix with all diagonal elements alike. To add two matrices of the same kind, we simply add the corresponding elements. Consider the set S of all n x m matrices (n and m fixed) and A and B are in S. From the properties of real numbers it's immediate that www.ping.be /~ping1339/matr.htm   (921 words)

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