MATRIX MULTIPLICATION (Part b) By: Shahrzad Abedi Professor: Dr. Haj Seyed Javadi MATRIX Multiplication • SIMD • MIMD – Multiprocessors – Multicomputers Chapter 7: Matrix Multiplication ‚ Parallel Computing :Theory and Practice‚ Michael J. Quinn 2 Matrix Multiplication Algorithms for Multiprocessors p1 p2 p3 p4 p1 p2 Chapter 7: Matrix Multiplication ‚ Parallel Computing :Theory and Practice‚ Michael J. Quinn p3 p4 3 Matrix Multiplication Algorithm for
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The nails and part of the hair are made of a durable‚ extensively cross-linked protein called hard ____. 2. A hair grows in a diagonal epithelial tube called a ____. 3. Coarse‚ pigmented hair is called ____‚ whereas most of the body hair of women and children is called ____. Most of the hair within this tube is called the root‚ but it has a dilation at its base called the ____‚ where it derives all its nutrition from blood capillaries. 4. The surface of a hair consists of scaly‚ overlapping
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applications. Notation: We use bold upper case letters to denote matrices and bold lower case letters to denote vectors. Furthermore‚ (·)−1 is reserved for the matrix inverse and (·)H for Hermitian transposition. The estimated value of a variable a is denoted by a. I denotes an identity matrix‚ diag(v) denotes a ˆ diagonal matrix with diagonal elements given by the vector v. · F denotes the Frobenius norm‚ E(·) is reserved for the mathematical expectation. Index Terms—OFDM‚ channel estimation
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of linear algebra or matrix theory. The introduction and development of the notation of matrix and system of linear algebra followed the development of determinant‚ which rose from the study of coefficient system of linear equations. Leibnitz‚ one of the founders of calculus used determinant in 1693 an Cramer presented his determinant - based formula system of solving linear equations (known as Cramer’s’ rule) in 1750. In 1848‚ James J Sylvester introduced the term “matrix” which was Latin word
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[pic] the best (minimum variance) linear (linear functions of the [pic]) unbiased estimator of [pic]is given by least squares estimator; that is‚ [pic]is the best linear unbiased estimator (BLUE) of [pic]. Proof: Let [pic]be any [pic]constant matrix and let [pic]; [pic] is a general linear function of [pic]‚ which we shall take as an estimator of [pic]. We must specify the elements of [pic]so that [pic]will be the best unbiased estimator of [pic]. Let [pic] Since [pic] is known‚ we must find
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Gaussian Elimination" (see Menu) Enter the matrix of coefficients and right-hand side vector You may edit the following statements or use the matrix and vector pallettes to enter new data ( see View‚ Palettes) > A:=<<4 | 2 | 3 | 2> ‚ <8 | 3 | -4 | 7> ‚ <4 | -6 | 2 | -5>>; > b:=<<15‚ 7‚ 6>>; Form the augmented matrix and solve The Maple routine GaussianElimination requires the augmented matrix A|b as input. In this worksheet this matrix is called Ab and is formed using <A|b> >
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13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Matrices—Two–Dimensional Arrays 13 16.1 Size of a matrix . . . . . . . . . . . . 14 16.2 Transpose of a matrix . . . . . . . . 14 16.3 Special Matrices . . . . . . . . . . . 14 16.4 The Identity Matrix . . . . . . . . . 14 16.5 Diagonal Matrices . . . . . . . . . . 15 16.6 Building Matrices . . . . . . . . . . . 15 16.7 Tabulating Functions . . . . . . . . . 15 16.8 Extracting Bits of Matrices
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1.1 SOLUTIONS Notes: The key exercises are 7 (or 11 or 12)‚ 19–22‚ and 25. For brevity‚ the symbols R1‚ R2‚…‚ stand for row 1 (or equation 1)‚ row 2 (or equation 2)‚ and so on. Additional notes are at the end of the section. 1. x1 + 5 x2 = 7 −2 x1 − 7 x2 = −5 1 −2 5 −7 7 −5 x1 + 5 x2 = 7 Replace R2 by R2 + (2)R1 and obtain: 3x2 = 9 x1 + 5 x2 = 7 x2 = 3 x1 1 0 1 0 1 0 5 3 5 1 0 1 7 9 7 3 −8 3 Scale R2 by 1/3: Replace R1 by R1
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Math 111 Homework 1 fall 2007 due 14/9 1. (1.2; 17) Determine the values of h such that the matrix is the augmented matrix of a system which admits a solution. 2 3 4 6 h 7 2. (1.2; 12) Find the general solutions of the system whose augmented matrix is 1 −7 0 6 5 0 0 1 −2 −3 −1 7 −4 2 7 1 −2 4 3. (1.3; 17) Let a1 = 4 ‚ a2 = −3 ‚ b = 1 . For what −2 7 h value(s) of h is b in the plane spanned by a1 and a2? 4. (1.4; 15) Let A = b1 2 −1 and b = . Show that the equation
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.. 15 Matrices .................................................................................................................................................... 27 Matrix Arithmetic & Operations .............................................................................................................. 33 Properties of Matrix Arithmetic and the Transpose ................................................................................. 45 Inverse Matrices and Elementary Matrices ............
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