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1. Matrix (mathematics) - Wikipedia, The Free Encyclopedia
In mathematics, a matrix (plural matrices) is a rectangular table of elements must be a square matrix (see Square matrices and related definitions below
http://en.wikipedia.org/wiki/Matrix_(mathematics)

Extractions: Jump to: navigation search In mathematics , a matrix (plural matrices ) is a rectangular table of elements (or entries ), which may be numbers or, more generally, any abstract quantities that can be added and multiplied . Matrices are used to describe linear equations , keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. Matrices are described by the field of matrix theory . Matrices can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra In this article, the entries of a matrix are real or complex numbers unless otherwise noted. Organization of a matrix The horizontal lines in a matrix are called rows and the vertical lines are called columns . A matrix with m rows and n columns is called an m -by- n matrix (written m n ) and m and n are called its dimensions . The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is commonly said that an

2. QuickMath Automatic Math Solutions
The matrices section of QuickMath allows you to perform arithmetic operations on For instance, when adding two matrices A and B, the element at row i,
http://www.quickmath.com/www02/pages/modules/matrices/index.shtml

3. Algebra II: Matrices - Math For Morons Like Us
On this page we hope to clear up problems that you might have with matrices. matrices are good things to have under control and know how to deal with,
http://library.thinkquest.org/20991/alg2/matrices.html

Extractions: On this page we hope to clear up problems that you might have with matrices. Matrices are good things to have under control and know how to deal with, because you will use them extensively in pre-calculus to solve systems of equations that have variables up the wazoo! (Like one we remember with seven equations in seven variables.) Addition and subtraction Quiz on Matrices To add matrices, we add the corresponding members. The matrices have to have the same dimensions. Example: Solution: Add the corresponding members. Subtraction of matrices is done in the same manner as addition. Always be aware of the negative signs and remember that a double negative is a positive! Back to Top You can multiply a matrix by another matrix or by a number. When you multiply a matrix by a number, multiply each member of the matrix by the number. To multiply a matrix by a matrix, the first matrix has to have the same number of columns as the rows in the second matrix. Examples: Solution: Multiply each member of the matrix by 2. Problem: Multiply the matrices shown below.

4. S.O.S. Math - Matrix Algebra
Introduction to Determinants Determinants of matrices of Higher Order Determinant and Inverse of matrices Application of Determinant to Systems
http://www.sosmath.com/matrix/matrix.html

5. Matrices And Determinants
DEFINITION Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the identical amount of
http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html

6. Matrices.net
Welcome to www.matrices.net! My name is Sara Howard, I m also known as matrices. This site is divided into three sections About Me which is a short
http://www.matrices.net/

Extractions: Welcome to www.matrices.net ! My name is Sara Howard, I'm also known as Matrices. This site is divided into three sections: "About Me" which is a short autobiography. "Art" the main focus of this site, which is a showcase of what I consider my artwork (including drawings, costuming and things like that). And "Other Stuff" which includes things like my journal, web cam, and links. Thank you for visiting, I hope you enjoy your stay. Go ahead and explore the site, and have fun! people have visited this site since July 21st 2001. Yeah, updates! All Fur Fun is the next convention I'm going to! Its April 18th - 20th in Spokane, WA. I'm already registered, so I'll be looking forward to seeing everyone there! My newest tutorial, the Paper Fox Army , a how-to on making origami foxes. Important Information and Pricing have been updated!

7. AMS Online Books/Letters On Matrices/COLL17
The 1934 classic Lectures on matrices by Wedderburn in scanned PDF.
http://www.ams.org/online_bks/coll17/

8. Matrices And Determinants
The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC.
http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants

Extractions: The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. However it was not until near the end of the 17 th Century that the ideas reappeared and development really got underway. It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around 300 BC contains the following problem:- There are two fields whose total area is square yards. One produces grain at the rate of of a bushel per square yard while the other produces grain at the rate of a bushel per square yard. If the total yield is bushels, what is the size of each field. The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. Indeed it is fair to say that the text Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. First a problem is set up which is similar to the Babylonian example given above:-

9. Matrices Worksheets, Determinants, Cramer's Rule, And More.
Determinants Mix of 2 x 2 and 3 x 3 matrices Determinants Calculate area of triangles Augmented matrices Write the Augmented Matrix and Solve
http://edhelper.com/Matrices.htm

10. An Introduction To MATRICES
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
http://home.scarlet.be/~ping1339/matr.htm

Extractions: Theorem 5 A matrix is an ordered set of numbers listed rectangular form. Example. Let A denote the matrix This matrix A has three rows and four columns. We say it is a 3 x 4 matrix. We denote the element on the second row and fourth column with a If a matrix A has n rows and n columns then we say it's a square matrix. In a square matrix the elements a i,i , with i = 1,2,3,... , are called diagonal elements.

11. Matrices
matrices. Notation of a Matrix and Operations with matrices Band matrices and Block matrices Determinants Four Subspaces of a Matrix
http://www.cs.ut.ee/~toomas_l/linalg/lin1/node11.html

12. Matrices
matrices are extremely handy for writing fast 3D programs. As you ll see they are just a 4x4 list of numbers, but they do have 2 very important properties
http://www.geocities.com/SiliconValley/2151/matrices.html

Extractions: Matrices Introduction Matrices are extremely handy for writing fast 3D programs. As you'll see they are just a 4x4 list of numbers, but they do have 2 very important properties: A tranformation is simply a way of taking a set of points and modifying them in some way to get a new set of points. For example, if the user moves 10 units forward in a certain direction then the net result is the same as if all objects in the world moved 10 units in the opposite direction. A Point in Space Modifying the Position of a Point the point Compare this to the artical on basic 3D math and you'll see that we are in fact taking the dot product of the two vectors. What we do above is mutiply each top item by the item under it and add the results up to get the answer.

13. Linear Equations, Matrices, Determinants
Take a fix matrix A. By crossing out, in a suitably way, some rows and some columns from A, we can construct many square matrices from A.
http://www.ping.be/~ping1339/stels2.htm

Extractions: Application about lines in a plane In previous articles, we have seen the fundamental properties of linear equation systems, matrices and determinants. In this part II, we bring these concepts together and we'll find many relations between these fundamentals. If the determinant of a square matrix is 0, we call this matrix singular otherwise, we call the matrix regular. Take a fix matrix A. By crossing out, in a suitably way, some rows and some columns from A, we can construct many square matrices from A. The number of rows of that matrix is called the rank of A. Replace each element of A with its own cofactor and transpose the result, then you have made the adjoint matrix of A. Theorem : When we multiply the elements of a row of a square matrix with the corresponding cofactors of another row, then the sum of these product is 0.

A library of Hadamard matrices maintained by NJA Sloane.

15. Rotation Matrix -- From Wolfram MathWorld
Orthogonal matrices have special properties which allow them to be manipulated be two orthogonal matrices. By the orthogonality condition, they satisfy
http://mathworld.wolfram.com/RotationMatrix.html

Extractions: Rotation Matrix When discussing a rotation , there are two possible conventions: rotation of the axes , and rotation of the object relative to fixed axes. In , consider the matrix that rotates a given vector by a counterclockwise angle in a fixed coordinate system. Then so This is the convention used by Mathematica command RotationMatrix theta On the other hand, consider the matrix that rotates the coordinate system through a counterclockwise angle . The coordinates of the fixed vector in the rotated coordinate system are now given by a rotation matrix which is the matrix transpose of the fixed-axis matrix and, as can be seen in the above diagram, is equivalent to rotating the vector by a counterclockwise angle of relative to a fixed set of axes, giving This is the convention commonly used in textbooks such as Arfken (1985, p. 195). In , coordinate system rotations of the x y -, and z -axes in a counterclockwise direction when looking towards the origin give the matrices (Goldstein 1980, pp. 146-147 and 608; Arfken 1985, pp. 199-200).

16. Homogeneous Transformation Matrices
Explicit ndimensional homogeneous matrices for projection, dilation, reflection, shear, strain, rotation and other familiar transformations.
http://www.silcom.com/~barnowl/HTransf.htm

Extractions: HOMOGENEOUS TRANSFORMATION MATRICES Daniel W. VanArsdale Vector (nonhomogeneous) methods are still being recommended to effect rotations and other linear transformations. Homogeneous matrices have the following advantages: simple explicit expressions exist for many familiar transformations including rotation these expressions are n-dimensional there is no need for auxiliary transformations, as in vector methods for rotation more general transformations can be represented (e.g. projections, translations) directions (ideal points) can be used as parameters of the transformation, or as inputs if nonsingular matrix T transforms point P by PT, then hyperplane h is transformed by T h the columns of T (as hyperplanes) generate the null space of T by intersections The expressions below use reduction to echelon form and Gram-Schmidt orthonormalization, both with slight modifications. They can be easily coded in any higher level language so that the same procedures generate transformations for any dimension. This article is at an undergraduate level, but the reader should have had some exposure to linear algebra and analytic projective geometry. This material is based on: Daniel VanArsdale, Homogeneous Transformation Matrices for Computer Graphics, , vol. 18, no. 2, pp. 177-191, 1994. Some

17. BLAST Substitution Matrices
The theory of amino acid substitution matrices is described in 1, and applied to DNA sequence comparison in 2. In general, different substitution
http://www.ncbi.nlm.nih.gov/blast/html/sub_matrix.html

Extractions: Query Length Substitution Matrix Gap Costs PAM-30 PAM-70 BLOSUM-80 BLOSUM-62 The raw score of an alignment is the sum of the scores for aligning pairs of residues and the scores for gaps. Gapped BLAST and PSI-BLAST use "affine gap costs" which charge the score -a for the existence of a gap, and the score -b for each residue in the gap. Thus a gap of k residues receives a total score of -(a+bk); specifically, a gap of length 1 receives the score -(a+b). To convert a raw score S into a normalized score S' expressed in bits, one uses the formula S' = (lambda*S - ln K)/(ln 2), where lambda and K are parameters dependent upon the scoring system (substitution matrix and gap costs) employed [7-9]. For determining S', the more important of these parameters is lambda. The "lambda ratio" quoted here is the ratio of the lambda for the given scoring system to that for one using the same substitution scores, but with infinite gap costs . This ratio indicates what proportion of information in an ungapped alignment must be sacrificed in the hope of improving its score through extension using gaps. We have found empirically that the most effective gap costs tend to be those with lambda ratios in the range 0.8 to 0.9. Altschul, S.F. (1993) "A protein alignment scoring system sensitive at all evolutionary distances." J. Mol. Evol. 36:290-300.

18. Tim Davis: UF Sparse Matrix Collection : Sparse Matrices From A Wide Range Of Ap
A collection of large sparse matrices from many scientific disciplines with links and software pieces to operate on matrix data structures.
http://www.cise.ufl.edu/research/sparse/matrices/

Extractions: From the abstract of the paper The University of Florida Sparse Matrix Collection: As of September 2007, it contains 1877 problems (some of which are sequences of dozens of matrices). The smallest is 5-by-5 with 19 nonzero entries. The largest has dimension 9.8 million, and the matrix with the most nonzeros has 99.2 million entries. The matrices are available in three formats: MATLAB mat-file, Rutherford-Boeing, and Matrix Market. The size of the collection in each format is about 9 GB. Note that the MATLAB mat-files can only be read by MATLAB 7.0 or later. This collection is managed by Tim Davis, but ``editors'' of other collections are attributed, via the Problem.ed field in each problem set. Problem.author is the matrix creator. Other collections are always welcome. Click here for a paper describing the collection (Jan. 2007).

19. Mathematics Reference: Rules For Matrices
Basic properties of matrices. A, B, and C are matrices,; O represents the zero matrix,; I represents the identity matrix,; r, s, and n are scalars.
http://www.alcyone.com/max/reference/maths/matrices.html

Extractions: MathRef Basic properties of matrices. Legend. Basic. -A == (-1) A equation 1 A - B == A + (-B) equation 2 1 A = A equation 3 A = O equation 4 A + O = O + A = A equation 5 I A = A I = A equation 6 A - A = O equation 7 Addition and scalar product. A + B = B + A equation 8 (A + B) + C = A + (B + C) equation 9 r (A + B) = r A + r B equation 10 r s ) A = r A + s A equation 11 r s ) A = r s A) equation 12 Matrix product. A == I equation 13 A == A A equation 14 A n = A A n equation 15 (A B) C = A (B C) equation 16 A (B + C) = A B + A C equation 17 (A + B) C = A C + B C equation 18 Transpose and inverse. I T = I equation 19 (A T T = A equation 20 (A + B) T = A T + B T equation 21 r A) T r A T equation 22 (A B) T = B T A T equation 23 I = I equation 24 A A = A A = I equation 25 (A B) = B A equation 26 (A T = (A T equation 27 Trace. tr (A + B) = tr A + tr B equation 28 tr ( r A) = r tr A equation 29 tr (A B) = tr (B A) equation 30 Determinant and adjoint.

20. Substitution Matrices
In aligning two protein sequences, some method must be used to score the alignment of one residue against another. Substitution matrices contain such values
http://arep.med.harvard.edu/seqanal/submatrix.html

Extractions: In aligning two protein sequences, some method must be used to score the alignment of one residue against another. Substitution matrices contain such values. The late Margaret Dayhoff was a pioneer in protein databasing and comparison. She and her coworkers developed a model of protein evolution which resulted in the development of a set of widely used substitution matrices . These are frequently called Dayhoff, MDM (Mutation Data Matrix), or PAM (Percent Accepted Mutation) matrices. Several later groups have attempted to extend Dayhoff's methodology or re-apply her analysis using later databases with more examples. Extensions Proteins 17:49 ) compared these two newer versions of the PAM matrices with Dayhoff's originals.

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