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.
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:
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.
The following matrix operations are available in MATLAB:
| + | addition |
| - | subtraction |
| * | multiplication |
| ^ | power |
| ' | transpose |
| \ | left division |
| / | right division |
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
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.
MATLAB is an expression language; the expressions you type are interpreted and evaluated. MATLAB statements are usually of the form
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.
Convenient matrix building functions are
| eye | identity matrix |
| zeros | matrix of zeros |
| ones | matrix of ones |
| diag | see below |
| triu | upper triangular part of a matrix |
| tril | lower triangular part of a matrix |
| rand | randomly generated matrix |
| hilb | Hilbert matrix |
| magic | magic square |
| toeplitz | see help toeplitz |
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.
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
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
| < | less than |
| > | greater than |
| <= | less than or equal |
| >= | greater than or equal |
| == | equal |
| ~= | not equal. |
Note that "=" is used in an assignment statement while "==" is used in a relation. Relations may be connected or quantified by the logical operators
| & | and |
| | | or |
| ~ | not. |
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
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
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.
Certain MATLAB functions operate essentially on scalars, but operate element-wise when applied to a matrix. The most common such functions are
| sin | asin | exp | abs | round |
| cos | acos | log (natural log) | sqrt | floor |
| tan | atan | rem (remainder) | sign | ceil |
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
| max | sum | median | any |
| min | prod | mean | all |
| sort | std |
For example, the maximum entry in a matrix A is given by max(max(A))
rather than max(A). Try it.