1. Accessing MATLAB

On most systems, after logging in one can enter MATLAB with the system command matlab and exit MATLAB with the command exit or quit. On a PC, for example, if properly installed, one may enter MATLAB with the command:

C> matlab

and exit it with the command:

>> quit

On systems permitting multiple processes, such as a Unix system, you will find it convenient, for reasons discussed in section 14, to keep both MATLAB and your local editor active. If you are working on a workstation which runs processes in multiple windows, you will want to keep MATLAB active in one window and your local editor active in another. You should consult your instructor or your local computer center for details of the local installation.

2. Entering matrices

MATLAB works with essentially only one kind of object---a rectangular numerical matrix with possibly complex entries; all variables represent matrices. In some situations, 1-by-1 matrices are interpreted as scalars and matrices with only one row or one column are interpreted as vectors.

Matrices can be introduced into MATLAB in several different ways:

• Entered by an explicit list of elements,
• Generated by built-in statements and functions,
• Created in M-files (see sections 12 and 14 below),
• Loaded from external data files (see User's Guide).

For example, either of the statements

A = [1 2 3; 4 5 6; 7 8 9]

and

A = [

1 2 3

4 5 6

7 8 9 ]

creates the obvious 3-by-3 matrix and assigns it to a variable A. Try it. The elements within a row of a matrix may be separated by commas as well as a blank.

When listing a number in exponential form (e.g. 2.34e-9), blank spaces must be avoided. Listing entries of a large matrix is best done in an M-file, where errors can be easily edited away (see sections 12 and 14).

The built-in functions rand, magic, and hilb, for example, provide an easy way to create matrices with which to experiment. The command rand(n) will create an n x n matrix with randomly generated entries distributed uniformly between 0 and 1, while rand(m,n) will create an m x n one. magic(n) will create an integral n x n matrix which is a magic square (rows and columns have common sum); hilb(n) will create the n x n Hilbert matrix, the king of ill-conditioned matrices (m and n denote, of course, positive integers). Matrices can also be generated with a for-loop (see section 6 below).

Individual matrix and vector entries can be referenced with indices inside parentheses in the usual manner. For example, A(2,3) denotes the entry in the second row, third column of matrix A and x(3) denotes the third coordinate of vector x. Try it. A matrix or a vector will only accept positive integers as indices.

3. Matrix operations, array operations

The following matrix operations are available in MATLAB:

These matrix operations apply, of course, to scalars (1-by-1 matrices) as well. If the sizes of the matrices are incompatible for the matrix operation, an error message will result, except in the case of scalar-matrix operations (for addition, subtraction, and division as well as for multiplication) in which case each entry of the matrix is operated on by the scalar.

The "matrix division" operations deserve special comment. If A is an invertible square matrix and b is a compatible column, resp. row, vector, then

x = A \ b is the solution of A * x = b and, resp.,

x = b / A is the solution of x * A = b.

In left division, if A is square, then it is factored using Gaussian elimination and these factors are used to solve A * x = b. If A is not square, it is factored using Householder orthogonalization with column pivoting and the factors are used to solve the under- or over-determined system in the least squares sense. Right division is defined in terms of left division by b / A = (A' \ b')'.

Array operations. The matrix operations of addition and subtraction already operate entry-wise but the other matrix operations given above do not---they are matrix operations. It is important to observe that these other operations, *, ^, \, and /, can be made to operate entry-wise by preceding them by a period. For example, either [1,2,3,4].*[1,2,3,4] or [1,2,3,4].\^2 will yield [1,4,9,16]. Try it. This is particularly useful when using Matlab graphics.

4. Statements, expressions, and variables; saving a session

MATLAB is an expression language; the expressions you type are interpreted and evaluated. MATLAB statements are usually of the form

variable = expression, or simply

expression

Expressions are usually composed from operators, functions, and variable names. Evaluation of the expression produces a matrix, which is then displayed on the screen and assigned to the variable for future use. If the variable name and = sign are omitted, a variable ans (for answer) is automatically created to which the result is assigned.

A statement is normally terminated with the carriage return. However, a statement can be continued to the next line with three or more periods followed by a carriage return. On the other hand, several statements can be placed on a single line if separated by commas or semicolons.

If the last character of a statement is a semicolon, the printing is suppressed, but the assignment is carried out. This is essential in suppressing unwanted printing of intermediate results.

MATLAB is case-sensitive in the names of commands, functions, and variables. For example, solveUT is not the same as solveut.

The command who will list the variables currently in the workspace. A variable can be cleared from the workspace with the command clear variablename. The command clear alone will clear all nonpermanent variables.

The permanent variable eps (epsilon) gives the machine precision---about 10^(-16) on most machines. It is useful in determining tolerences for convergence of iterative processes.

A runaway display or computation can be stopped on most machines without leaving MATLAB with CTRL-C (CTRL-BREAK on a PC).

Saving a session. When one logs out or exits MATLAB all variables are lost. However, invoking the command save before exiting causes all variables to be written to a non-human-readable diskfile named matlab.mat. When one later reenters MATLAB, the command load will restore the workspace to its former state.

5. Matrix building functions

Convenient matrix building functions are

For example, zeros(m,n) produces an m-by-n matrix of zeros and zeros(n) produces an n-by-n one; if A is a matrix, then zeros(A) produces a matrix of zeros of the same size as A.

If x is a vector, diag(x) is the diagonal matrix with x down the diagonal; if A is a square matrix, then diag(A) is a vector consisting of the diagonal of A. What is diag(diag(A))? Try it.

Matrices can be built from blocks. For example, if A is a 3-by-3 matrix, then

B = [A, zeros(3,2); zeros(2,3), eye(2)]

will build a certain 5-by-5 matrix. Try it.

6. For, while, if --- and relations

In their basic forms, these MATLAB flow control statements operate like those in most computer languages.

For. For example, for a given n, the statement

x = []; for i = 1:n, x=[x,i^2], end

or

x = [];

for i = 1:n

x = [x,i^2]

end

will produce a certain n-vector and the statement

x = []; for i = n:-1:1, x=[x,i^2], end

will produce the same vector in reverse order. Try them. Note that a matrix may be empty (such as x = []). The statements

for i = 1:m

for j = 1:n

H(i, j) = 1/(i+j-1);

end

end

H

will produce and print to the screen the m-by-n hilbert matrix. The semicolon on the inner statement suppresses printing of unwanted intermediate results while the last H displays the final result.

While. The general form of a while loop is

while relation

statements

end

The statements will be repeatedly executed as long as the relation remains true. For example, for a given number a, the following will compute and display the smallest nonnegative integer n such that 2^n>= a:

n = 0;

while 2^n < a

n = n + 1;

end

n

If. The general form of a simple if statement is

if relation

statements

end

The statements will be executed only if the relation is true. Multiple branching is also possible, as is illustrated by

if n < 0

parity = 0;

elseif rem(n,2) == 0

parity = 2;

else

parity = 1;

end

In two-way branching the elseif portion would, of course, be omitted.

Relations. The relational operators in MATLAB are

Note that "=" is used in an assignment statement while "==" is used in a relation. Relations may be connected or quantified by the logical operators

When applied to scalars, a relation is actually the scalar 1 or 0 depending on whether the relation is true or false. Try 3 < 5, 3 > 5, 3 == 5, and 3 == 3. When applied to matrices of the same size, a relation is a matrix of 0's and 1's giving the value of the relation between corresponding entries. Try a = rand(5), b = triu(a), a == b.

A relation between matrices is interpreted by while and if to be true if each entry of the relation matrix is nonzero. Hence, if you wish to execute statement when matrices A and B are equal you could type

if A == B

statement

end

but if you wish to execute statement when A and B are not equal, you would type

if any(any(A ~= B))

statement

end

or, more simply,

if A == B else

statement

end

Note that the seemingly obvious

if A ~= B, statement, end

will not give what is intended since statement would execute only if each of the corresponding entries of A and B differ. The functions any and all can be creatively used to reduce matrix relations to vectors or scalars. Two any's are required above since any is a vector operator (see section 8).

The for statement permits any matrix to be used instead of 1:n. See the User's Guide for details of how this feature expands the power of the for statement.

7. Scalar functions

Certain MATLAB functions operate essentially on scalars, but operate element-wise when applied to a matrix. The most common such functions are

8. Vector functions

Other MATLAB functions operate essentially on a vector (row or column), but act on an m-by-n matrix (m >= 2) in a column-by-column fashion to produce a row vector containing the results of their application to each column. Row-by-row action can be obtained by using the transpose; for example, mean(A')'. A few of these functions are

For example, the maximum entry in a matrix A is given by max(max(A)) rather than max(A). Try it.

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