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Lecture_03_Matrix_Math_Gauss_Elimination-1

# Lecture_03_Matrix_Math_Gauss_Elimination-1 - EGR 102...

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EGR 102 Introduction to Engineering Modeling atrix Math Matrix Math Chapter 8.1 Gauss Elimination Chapter 8.1, 9.1-9.3 EGR 102 Lecture 3 1 Figures from: “Applied Numerical Methods with MATLAB,” Steven Chapra, McGraw Hill

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MATLAB: Matrix Laboratory Images, movies, sound and variables all are matrices in MATLAB Sound: load chirp sound(y) Image: load clown image(X), colormap(map) Animations (movies): z = peaks; surf(z); axis tight set(gca,'nextplot','replacechildren'); for j = 1:20 surf(sin(2*pi*j/20)*z,z) f(j) = getframe; EGR 102 Lecture 3 2 end movie(f,50)
MATLAB ± MATLAB is both a software tool and a computer language. ± A variety of numerical functions, symbolic computations, and visualization tools have been added to the matrix manipulations. ± Other languages & libraries include: ± FORTRAN (IMSL), C++,… EGR 102 Lecture 3 3

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Numerical Methods … • Solving a system of algebraic equations / inverting atrices matrices • Finding roots of equations olving matrices (linear algebra) • Solving matrices (linear algebra) • Regression / Curve-fitting umerical Integration • Numerical Integration • Solving ODE’s (Ordinary Differential Equations) ore advanced topics ( e g solving PDEs) EGR 102 Lecture 3 4 • More advanced topics ( e.g., solving PDEs) …
Finite Element Analysis illions of Millions of simultaneous equations EGR 102 Lecture 3 5

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Matrix Math Review ± Matrices used to: escribe linear equations ± Describe linear equations ± Track coefficients of linear transformations ecord data that depend on multiple parameters ± Record data that depend on multiple parameters ± Matrices can be added, multiplied, and ecomposed in various ways decomposed in various ways EGR 102 Lecture 3 6
Matrix Math Review ± The horizontal lines in a matrix are called rows ± The vertical lines are called columns . ± A matrix with m rows & n columns is called an m-by-n matrix ± mand nare 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 m-by-n matrix has an order (size) of m × n. = 2 1 0 8 7 6 5 4 3 2 1 A EGR 102 Lecture 3 7 12 11 10 9

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Matrix Math Review ± Almost always capital letters denote matrices with the corresponding lower-case letters with two indices representing the entries. or example the entry of a matrix at lies in the - row and the - ± For example, the entry of a matrix A that lies in the i th row and the j th column is written as a i,j and called the i,j entry or (i,j)-th entry of A . ± Alternative notations for that entry are A [i,j] or A . The row is always oted first then the column noted first, then the column. 4 3 2 1 ? = 12 11 10 9 8 7 6 5 A a 2,3 = ? EGR 102 Lecture 3 8
Vectors ± A matrix where one of the dimensions equals 1 is often called a vector ± An m × 1 matrix (1 column & m rows) is called a column vector 1 atrix (1 row & lumns) is called a w ector ± A 1 × n matrix (1 row & n columns) is called a row vector { } 4 3 2 1 = A EGR 102 Lecture 3 9

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Special Matrices Matrices where m=n are called square matrices.
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Lecture_03_Matrix_Math_Gauss_Elimination-1 - EGR 102...

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