Slightly revised version of Scilab and Matrices by David Arnold, College of the Redwoods, August 17, 1998.

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1 Scilab and Matrices Slightly revised version of Scilab and Matrices by David Arnold, College of the Redwoods, August 7, 998. In this exercise you will learn how to enter and edit matrices in Scilab. You will also experiment with some of Scilab s built-in matrix builders, and you will learn how to build matrices from vectors and blocks of matrices. Matrices Entering Matrices Entering matrices is easy in Scilab. Enter the following command at the Scilab prompt -- >. --> A=[ 2 3;4 5 6;7 8 9] Delimit (separate) the entries in a row with commas or spaces and delimit the rows with semicolons. --> A=[,2,3;4,5,6;7,8,9] Special Matrices in Scilab Scilab has a number of built-in routines to help create special matrices. You can create a matrix of zeros of any size. (Type help zeros at the prompt for more information about zero matrices.) --> A=zeros(5,5) --> B=zeros(3,5) You can create matrices of ones in a similar manner. --> C=ones(2,) You can create an identity matrix ( s on the diagonal, s everywhere else) by typing the following command. --> D=eye(5,5)

2 When performing simulations in Scilab it is useful to build matrices of random numbers. You can create a matrix of uniformly distributed random numbers [,) with the following commands. (Type "help rand" for more information.) --> A=rand(6,6) --> B=rand(5,3) Multiplication of a matrix by a scalar multiplies every element of the matrix by the scalar, just as with vectors. --> A=*rand(5,5)-3 Use Scilab s help command to find out the difference between fix, round, ceil and floor. Then execute the commands: --> B=round(A) --> B=ceil(A) --> B=floor(A) --> B=fix(A) and explain the difference in each result. If rand produces a random number in the interval [,), then 5*rand should produce a random number in the interval [,5). --> C=5*rand(6,6) If 5*rand produces a random number in the interval [,5), then 5*rand-2 should produce a random number in the interval [-2,3). --> C=5*rand(6,6)-2 And the floor command should round down to the nearest integer. --> C=floor(5*rand(6,6) -2) Explain how you can produce a random 6 by 6 matrix, containing integers randomly selected from amongst the integers (-4,-3,-2,-,,,2,3,4). Other types of diagonal matrices are possible with Scilab s diag command.

3 --> E=diag([,2,3,4,5]) --> F=diag([,2,3,4,5], i) -->//shift diagonal up (positive i) or down (negative i) --> G=diag(:5,) The Transpose Operator The transpose operator (single apostrophe) plays the same role with matrices as it does with vectors. Rows are changed to columns and columns are changed to rows. --> J=[ 2 3;4 5 6;7 8 9] --> J Suppressing Output Remember, appending a semicolon to any Scilab command will suppress the output. This is useful if the result is large and you have no desire to view it. --> K=rand(,); Indexing Matrices The following notation is used to denote a matrix with 3 rows and 3 columns. a a 2 a 3 a 2 a 22 a 32 a 3 a 23 a 33 Note that a 2 refers to the entry in row, column 2. In general, a ij refers to the entry in row i, column j. Scilab uses similar index notation to identify the elements of a matrix. --> A(,2) --> A(3,4) In general, A(i,j) refers to the element in row i, column j of matrix A.

4 You can easily change an entry in a matrix. --> A(3,3)=234 Advanced Indexing Techniques When indexing a matrix, your subscripts can also be vectors. This is a powerful tool which allows you to easily select a submatrix of a given matrix. --> A=round(*rand(6,6)) --> A([,2],[3,4,5]) The notation A([,2],[3,4,5]) references the submatrix formed using the elements that appear in rows and 2 and in columns 3, 4 and 5 of the matrix A. --> A([,3,5],[,2,3,4,5,6]) produces a submatrix composed of rows, 3 and 5 of matrix A. If you recall, the notation :6 represents the vector [,2,3,4,5,6], so that A(,:6) is equivalent to A(,[,2,3,4,5,6]). --> A(,:6) Using a colon by itself in place of a subscript denotes all of the corresponding row or column. Thus --> A(:,) produces the first column of the matrix A (all rows, first column) and --> A(3,:) produces the third row of the matrix A. In a sense, you can read the notation A(3,:) as: Third row, every column. --> A(:3,:) produces a submatrix composed of the first three rows of the matrix A, while the command

5 --> A(:,:2:6) produces a submatrix composed of columns, 3 and 5 of matrix A. Building Matrices Scilab allows you to build complex matrices out of vectors and matrices. Building Matrices From Vectors Create three row vectors with the following commands. --> vl=:3 --> v2=4:6 --> v3=7:9 --> M=[v;v2;v3] builds a matrix with the vectors v, v2 and v3 each forming a row of the matrix M. --> N=[v, v2, v3] produces a significantly different but logical result. Change the vectors v, v2 and v3 into column vectors with the transpose operator. --> v=vl' --> v2=v2' --> v3=v3' --> P=[v,v2,v3] builds a matrix with the vectors v, v2 and v3 each forming a column of the matrix P. Of course, you could have achieved the same result by taking the transpose of the matrix M.

6 --> M' The Dimensions Need To Match When building matrices, be sure that each row and column has the same number of elements. For example, the following sequence of commands will produce an error. --> w=:3, w2=4:6, w3=7: --> Q=[wl;w2;w3] Building Matrices from Other Matrices It is a simple matter to augment a row or column vector to a matrix. Remember, each row and column of a matrix must have the same number of elements. Thus --> A=[ 2 3 4; ;9 2] --> b=[ ] --> M=[A b] is valid, but --> M=[A;b] is not. You can also augment two or more matrices. Remember, each row and column of a matrix must have the same number of elements. Thus --> A=rand(3,3),B=ones(3,4) --> M=[A B] is valid, but --> N=[A;B] is not. Your Imagination Is Your Only Limit Scilab s matrix building capability is extremely flexible. Here is an interesting example.

7 --> A=zeros(3,3),B=ones(3,3),C=2*ones(3,3),D=3*ones(3,3) --> M=[A B;C D] Here is something called a Vandermonde matrix. --> x=[ ] --> N=[x.^ x x.^2 x.^3 x.^4] And here is one final piece of chicanery. --> B=zeros(8,8) --> B=(:3,:3)=[ 2 3;4 5 6;7 8 9] --> B(4:8,4:8)=testmatrix('magi',5) Homework Write the commands on a piece of paper that will perform the tasks required in each of the following xercises. Check your solution at the computer. ) What command will produce each of the following matrices? a) b) A = [ ] B = c)

8 C = ) Explain how you can produce a 5 by 3 matrix whose entries are random integers between a) and 6. b) -3 and 3. 3) What is the easiest way to create each of the following matrices? a) A = b) B = 3 2 4) Enter A = testmatrix('magi',5). What is the result of each of the following commands? a) A(3,:) b) A(:,4) c) A(:2,:2:5) 5) What is an easy way to create the following Vandermonde matrix? V =

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