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Chapter 8. Matrices

# Chapter 8. Matrices - 8 Matrices 8.1 Setting up Matrices...

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0-8493-????-?/00/\$0.00+\$.50 © 2000 by CRC Press LLC © 2001 by CRC Press LLC 8 Matrices 8.1 Setting up Matrices DEFINITION A matrix is a collection of numbers arranged in a two-dimen- sional (2-D) array structure. Each element of the matrix, call it M i,j , occupies the i th row and j th column. (8.1) We say that M is an ( m n ) matrix, which means that it has m rows and n columns. If m = n , we call the matrix square. If m = 1, the matrix is a row vec- tor; and if n = 1, the matrix is a column vector. 8.1.1 Creating Matrices in MATLAB 8.1.1.1 Entering the Elements In this method, the different elements of the matrix are keyed in; for example: M=[1 3 5 7 11; 13 17 19 23 29; 31 37 41 47 53] gives M = 1 3571 1 13 17 19 23 29 31 37 41 47 53 M = MMM M M M n n mmm m n 11 12 13 1 21 22 23 2 123 L L O M L

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© 2001 by CRC Press LLC To fnd the size oF the matrix (i.e., the number oF rows and columns), enter: size(M) gives ans = 35 To view a particular element, For example, the (2, 4) element, enter: M(2,4) gives ans = 23 To view a particular row such as the 3 rd row, enter: M(3,:) gives ans = 31 37 41 47 53 To view a particular column such as the 4 th column, enter: M(:,4) gives ans = 7 23 47 IF we wanted to construct a submatrix oF the original matrix, For example, one that includes the block From the 2 nd to 3 rd row (included) and From the 2 nd column to the 4 th column (included), enter: M(2:3,2:4)
© 2001 by CRC Press LLC gives ans = 17 19 23 37 41 47 8.1.1.2 Retrieving Special Matrices from the MATLAB Library MATLAB has some commonly used specialized matrices in its library that can be called as needed. For example: • The matrix of size ( m n ) with all elements being zero is M=zeros(m,n); For example: M=zeros(3,4) gives M = 0000 • The matrix of size ( m n ) with all elements equal to 1 is N=ones(m,n) : For example: N=ones(4,3) produces N = 111 • The matrix of size ( n n ) with only the diagonal elements equal to one, otherwise zero, is P=eye(n,n) : For example:

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© 2001 by CRC Press LLC P=eye(4,4) gives P = 1000 0100 0010 0001 • The matrix of size ( n n ) with elements randomly chosen from the interval [0, 1], such as: Q=rand(4,4) gives, in one instance: Q = • We can select to extract the upper triangular part of the Q matrix, but assign to all the lower triangle elements the value zero: upQ=triu(Q) produces upQ = or extract the lower triangular part of the Q matrix, but assign to all the upper triangle elements the value zero: loQ=tril(Q) produces loQ = 0.9708 0.4983 0.9601 0.2679 0.9901 0.2140 0.7266 0.4399 0.7889 0.6435 0.4120 0.9334 0.4387 0.3200 0.7446 0.6833 0.9708 0.4983 0.9601 0.2679 0 0.2140 0.7266 0.4399 0 0 0.4120 0.9334 0 0 0 0.6833
© 2001 by CRC Press LLC • The single quotation mark (‘) after the name of a matrix changes the matrix rows into becoming its columns, and vice versa, if the elements are all real. If the matrix has complex numbers as ele- ments, it also takes their complex conjugate in addition to the transposition. • Other specialized matrices, including the whole family of sparse matrices, are also included in the MATLAB library. You can Fnd more information about them in the help documentation.

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Chapter 8. Matrices - 8 Matrices 8.1 Setting up Matrices...

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