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23. An Introduction To MATRICES An introduction to matrices. transpose. Then a i,j = a j,i , for all i and j. The sum of matrices of the same kind. Sum of matrices. To http://www.ping.be/~ping1339/matr.htm

An introduction to MATRICES Definitions Matrix Square matrix Diagonal matrix ... Theorem 5 Definitions Matrix 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 Square matrix 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. Remark. There is no difference between a 1 x 1 matrix and an ordenary number. Diagonal matrix A diagonal matrix is a square matrix with all de non-diagonal elements 0. The diagonal matrix is completely denoted by the diagonal elements. Example. [7 0] [0 5 0] [0 6] The matrix is denoted by diag(7 , 5 , 6) Row matrix A matrix with one row is called a row matrix Column matrix A matrix with one column is called a column matrix Matrices of the same kind Matrix A and B are of the same kind if and only if A has as many rows as B and A has as many columns as B The tranpose of a matrix The n x m matrix A' is the transpose of the m x n matrix A if and only if The ith row of A = the ith column of A' for (i = 1,2,3,..n)

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25. Linear Equations, Matrices, Determinants Linear equations, matrices, determinants. fundamentals. Rank of a matrix and inverse of a matrix. Singular and regular matrices. If http://www.ping.be/~ping1339/stels2.htm

Linear equations, matrices, determinants Introduction Rank of a matrix and inverse of a matrix Singular and regular matrices Rank of a matrix ... Application about lines in a plane Introduction 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. Rank of a matrix and inverse of a matrix Singular and regular matrices If the determinant of a square matrix is 0, we call this matrix singular otherwise, we call the matrix regular. Rank of a matrix 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. Doing this, search now the biggest regular square matrix. The number of rows of that matrix is called the rank of A. Adjoint matrix of a square matrix A Replace each element of A with its own cofactor and transpose the result, then you have made the adjoint matrix of A. Cofactors property 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.

26. Matrices And Determinants matrices and determinants. Algebra index. History Topics Index. The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants

Matrices and determinants Algebra index History Topics Index 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:-

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28. Matrices Watch us work! Home Free Study Aids Study Guides Math matrices Table of Contents. Summary. Terms. matrices. Problems. Multiplication. Problems. http://www.sparknotes.com/math/algebra2/matrices/

Advanced Search FAQ document.write("My Preferences"); My Preferences We want your feedback! Please let us know if you have any document.write("comments, "); document.write("requests, "); document.write("or if you think you've found an "); document.write("error."); comments requests , or if you think you've found an error Watch us work! Home Free Study Aids ... Matrices Table of Contents - Navigate Here - Summary Terms Matrices >Problems Multiplication >Problems Identity Matrix >Problems Row Reduction >Problems Inverses >Problems Summary Terms Matrices >Problems ... Geometry SparkNote by Lara Buchak How do I cite this study guide? - Navigate Here - Summary Terms Matrices >Problems Multiplication >Problems Identity Matrix >Problems Row Reduction >Problems Inverses >Problems Terms and Conditions Contact Information

29. NIWMS-2000 - Ninth International Workshop On Matrices And Statistics Ninth International Workshop on matrices and Statistics, in celebration of C.R. Rao's 80th Birthday. Hyderabad, India; 913 December 2000. http://eos.ect.uni-bonn.de/HYD2000.htm

30. Weight Matrices For Sequence Similarity Scoring Weight matrices for Sequence Similarity Scoring. Version 2.0 May 1996. David Weight matrices for Sequence Similarity Scoring. Outline Objective http://www.techfak.uni-bielefeld.de/bcd/Curric/PrwAli/nodeD.html

Weight Matrices for Sequence Similarity Scoring Version 2.0 May 1996 David Wheeler , Ph.D. Department of Cell Biology, Baylor College of Medicine Houston, Texas E-mail: wheeler@bcm.tmc.edu Table of Contents Weight matrices for sequence similarity scoring Importance of scoring matrices Examples of matrices Log odds matrices ... Other specialized scoring matrices Weight Matrices for Sequence Similarity Scoring Outline: Objective: Overview of methods and theories that underlie the construction of scoring matrices. Examples of weight matrices for nucleotide and amino acid scoring. Transition probability matrix: PAM Construction Properties Sources of error BLOSUM matrix Construction Sources of error Practical aspects Other refinements to transition probability matrices. Reading: D.G. George, W. C. Barker and L. T. Hunt. (1990). Mutation Data Matrix and Its Uses. in Methods in Enzymology vol 183; R.F. Doolittle, ed. pp. 333-351. Academic Press, Inc. New York. M.O. Dayhoff (1978) Atlas of Protein Sequence and Structure (Natl. Biomed. Res. Found., Washington), Vol. 5, Suppl. 3, pp. 345-352. S.F. Altschul (1991). Amino acid substitution matrices from an information theoretic perspective. J. Mol. Biol. 219 555-565.

31. PAM Matrices PAM matrices. There are several common ways in which weights can be applied for amino acid differences. This results in a family of scoring matrices. http://helix.biology.mcmaster.ca/721/distance/node9.html

Next: BLOSUM Matrices Up: Amino acid distance Previous: Amino acid distance PAM Matrices There are several common ways in which weights can be applied for amino acid differences. Karlin and Ghandour (1985, PNAS 82:8597) proposed a method of weights based on chemical, functional, charge and structural properties of the amino acids. Similarly Doolittle proposed weights based on the structural similarities and the ease of genetic interchange (Feng et al ., 1985 J. Mol. Evol. 21: 112). However, by far the most common and most famous way to assign weights is to use Dayhoff's PAM250 matrix. This is a matrix of weights that is derived from how often different amino acids replace other amino acids in evolution (see M.O. Dayhoff, ed., 1978, Atlas of Protein Sequence and Structure, Vol5). This was based on a data base of 1,572 changes in 71 groups of closely related proteins appearing in earlier volumes of this amazing predecessor to electronic databases. PAM stands for percent accepted mutations and these were inferred from the types of changes observed in these proteins. Every change was tabulated and entered in a matrix enumerating all possible amino acid changes. In addition to these counts of accepted point mutations an idea of the relative mutability of different amino acids were calculated. The information about the individual kinds of mutations and about the relative mutability of the amino acids can be combined into one distance-dependent "mutation probability matrix". The elements of this matrix give the probability that the amino acid in one column will be replaced by the amino acid in some row after a given evolutionary interval. For example, a matrix with an evolutionary distance of PAMs would have ones on the main diagonal and zeros elsewhere. A matrix with an evolutionary distance of

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33. BLOSUM Matrices BLOSUM matrices. The BLOSUM matrices originate with a paper by Henikoff and Henikoff (1992; PNAS 891091510919). Their idea was http://helix.biology.mcmaster.ca/721/distance/node10.html

Next: GAP WEIGHTING Up: Amino acid distance Previous: PAM Matrices BLOSUM Matrices The BLOSUM matrices originate with a paper by Henikoff and Henikoff (1992; PNAS 89:10915-10919). Their idea was to get a better measure of differences between two proteins specifically for more distantly related proteins. While this bias limits the usefulness of BLOSUM matrices for some purposes, for other programs such as FASTA, BLAST, etc. it should do substantially better. This is because the need for an accurate measure of distance is not as great when peptides are more closely related. They use the BLOCKS database to search for differences among sequences but only among the very conserved regions of a protein family. Hence the term BLOSUM is from BLOcks SUbstitution Matrix. They first collect all of the sequences in the BLOCKS database and then for each one they sum the number of amino acids in each site to get a frequency table ( ) of how often different pairs of amino acids are found together in these conserved regions. Hence the observed frequency of occurrence of one amino acid is Given pairs should occur with frequencies and The odds matrix is . Generally 's are taken of this matrix to give a or lod matrix such that Hence if the observed number of differences between a pair of amino acids is equal to the expected number than . If the observed is less than expected then and if the observed is greater than expected All of this gives the BLOSUM matrix. Different levels of the BLOSUM matrix can be created by differentially weighting the degree of similarity between sequences. For example, a BLOSUM62 matrix is calculated from protein blocks such that if two sequences are more than 62% identical, then the contribution of these sequences is weighted to sum to one. In this way the contributions of multiple entries of closely related sequences is reduced.

34. Some Codes Examples for q up to 9. http://www.mathi.uni-heidelberg.de/~yves/Matritzen/Codes/CodeMatIndex.html

Check- or generator matrices of some linear codes I give here check- or generator matrices of some linear codes. Most of these codes were found by computer search. Codes cited in older papers, whose parameters are implied by other codes given here, are set italic. Look at caps for codes with distance 4. Check- or generator matrices of some linear binary codes Check- or generator matrices of some linear ternary codes. Check- or generator matrices of some linear quaternary codes. Check- or generator matrices of some linear codes for q=5. Check- or generator matrices of some linear codes for q=7. Check- or generator matrices of some linear codes for q=8. Check- or generator matrices of some linear codes for q=9. home

35. MATRICES Translate this page matrices. El número total de elementos de una matriz Am×n es m·n En matemáticas, tanto las Listas como las Tablas reciben el nombre genérico de matrices. http://personal5.iddeo.es/ztt/Tem/T6_Matrices.htm

MATRICES Las matrices aparecen por primera vez hacia el año 1850, introducidas por J.J. Sylvester El desarrollo inicial de la teoría se debe al matemático W.R. Hamilton en 1853 En 1858, A. Cayley introduce la notación matricial como una forma abreviada de escribir un sistema de m ecuaciones lineales con n incógnitas. Las matrices se utilizan en el cálculo numérico, en la resolución de sistemas de ecuaciones lineales, de las ecuaciones diferenciales y de las derivadas parciales. Además de su utilidad para el estudio de sistemas de ecuaciones lineales, las matrices aparecen de forma natural en geometría, estadística, economía, informática, física, etc... La utilización de matrices (arrays) constituye actualmente una parte esencial dn los lenguajes de programación, ya que la mayoría de los datos se introducen en los ordenadores como tablas organizadas en filas y columnas : hojas de cálculo, bases de datos,... CONCEPTO DE MATRIZ Una matriz es un conjunto de elementos de cualquier naturaleza aunque, en general, suelen ser números ordenados en filas y columnas. Se llama matriz de orden "m

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

37. PRESENTACION matrices, troqueles sistemas de ensamblaje de conjuntos de chapa para el autom³vil. http://www.inmapa.com/presentacion.htm

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Getting Started Matrices and Arrays Matrices and Magic Squares Enter matrices, perform matrix operations, and access matrix elements. Expressions Work with variables, numbers, operators, functions, and expressions. Working with Matrices Generate matrices, load matrices, create matrices from M-files and concatenation, and delete matrix rows and columns. More About Matrices and Arrays Use matrices for linear algebra, work with arrays, multivariate data, scalar expansion, and logical subscripting, and use the find function. Controlling Command Window Input and Output Change output format, suppress output, enter long lines, and edit at the command line. Starting and Quitting MATLAB Matrices and Magic Squares Trademarks

39. Departamento De Matemáticas -- UCA Teor­a, problemas y soluciones (pdf). Universidad de C¡diz. http://matematicas.uca.es/empres/analisis/

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40. Linear Algebra :: Functions -- Categorical List (MATLAB Functions) Arrays and matrices. cumprod, Cumulative product. cumsum, Cumulative sum. diag, Diagonal matrices and diagonals of matrix. dot, Vector dot product. end, Last index. http://www.mathworks.com/access/helpdesk/help/techdoc/ref/func_b11.html

MATLAB Function Reference Linear Algebra Matrix Analysis Linear Equations Eigenvalues and Singular Values Matrix Logarithms and Exponentials ... Factorization Matrix Analysis cond Condition number with respect to inversion condeig Condition number with respect to eigenvalues det Determinant norm Matrix or vector norm normest Estimate matrix 2-norm null Null space orth Orthogonalization rank Matrix rank rcond Matrix reciprocal condition number estimate rref Reduced row echelon form subspace Angle between two subspaces trace Sum of diagonal elements Linear Equations Linear equation solution chol Cholesky factorization cholinc Incomplete Cholesky factorization cond Condition number with respect to inversion condest 1-norm condition number estimate funm Evaluate general matrix function inv Matrix inverse linsolve Solve linear systems of equations lscov Least squares solution in presence of known covariance lsqnonneg Nonnegative least squares lu LU matrix factorization luinc Incomplete LU factorization pinv Moore-Penrose pseudoinverse of matrix qr Orthogonal-triangular decomposition rcond Matrix reciprocal condition number estimate Eigenvalues and Singular Values balance Improve accuracy of computed eigenvalues Convert complex diagonal form to real block diagonal form condeig Condition number with respect to eigenvalues eig Eigenvalues and eigenvectors eigs Eigenvalues and eigenvectors of sparse matrix gsvd Generalized singular value decomposition hess Hessenberg form of matrix poly Polynomial with specified roots polyeig Polynomial eigenvalue problem qz QZ factorization for generalized eigenvalues

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