Matlab Arrays

From Sutherland_wiki
Jump to: navigation, search


Array Basics

An array is an n-dimensional collection of numbers. Matrices are 2-dimensional arrays, and vectors are 1-dimensional arrays.

  • Vectors can be either a row or column vector:

\( \quad a= \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{bmatrix} \quad b = \begin{bmatrix} b_1 & b_2 & \cdots & b_n \end{bmatrix} \quad c = \begin{bmatrix} 1 \\ 4 \\ 3.2 \\ \pi \end{bmatrix} \)

We refer to the elements of an array by their position in the array. For example, the third element in \(c\) is 3.2, and the second element in \(a\) is \(a_2\).
  • Matrices are two-dimensional arrays, whose elements are referred to by the row and column that they belong to. For example,

\( \quad A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix} \)

is a matrix with three rows and two columns. We refer to this as a \(\scriptstyle 3 \times 2\) matrix. The first index refers to the row and the second index refers to the column.

Transposing Arrays

Occasionally we want to transpose an array. This is done by exchanging rows and columns. For example,

\( A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \\ 10 & 12 \end{bmatrix} \quad \Rightarrow \quad A^\mathsf{T} = \begin{bmatrix} 2 & 6 & 10 \\ 4 & 8 & 12 \end{bmatrix} \)

Here the \(^\mathsf{T}\) superscript indicates transpose. Note that if we transpose an array twice, we recover the original array. In other words, \(\left(A^\mathsf{T}\right)^\mathsf{T} = A\).

In MATLAB, the transpose operator is given by an apostrophe, '. For example,

   A = [2 4; 6 8; 10 12 ];  % create the "A" matrix.
   Atrans = A';             % create a new matrix holding the transpose of "A"

The first line of code here creates the matrix and the second stores its transpose.

Creating Arrays in Matlab

In this section we will discuss several ways of creating arrays (matrices and vectors) in MATLAB.

Manual Creation

Often we need to create small arrays with only a few numbers. In this case, it is easiest to create the vector in one Matlab statement. The following table illustrates how to create vectors in Matlab.

Vector type Example Matlab Code
Column vector \( a = \left[\begin{smallmatrix} 2 \\ 4 \\ 6 \\ 8 \end{smallmatrix} \right] \)
 a = [ 2; 4; 6; 8; ];  % create a column vector
 a = [ 2 4 6 8 ]';     % create a row vector and then
                       % transpose it to produce a column vector.
 a = [ 2
       4
       6
       8 ];  % creates a column vector.
Row vector \( b = \left[ \begin{smallmatrix} 1 & 3 & 5 & 7 \end{smallmatrix} \right] \)
  b = [ 1 3 5 7 ];
  b = [ 1, 3, 5, 7 ];
  b = [ 1; 3; 5; 7 ]';  % transpose a column vector.

Note that the semicolon (;) identifies the end of a row in the matrix. Commas may be used to separate columns in the matrix, but they are not necessary.

Matrices may be created directly in much the same way as vectors are. As an example, let's build the matrix \[ A = \left[\begin{smallmatrix} 2 & 4 \\ 6 & 8 \\ 10 & 12 \end{smallmatrix}\right]. \] This may be done in several ways.

  A = [ 2 4; 6 8; 10 12 ];
  A = [ 2 4
        6 8
        10 12 ];

Note that a semicolon (;) separates each row of the matrix while spaces separate each column.

We may also create a matrix from vectors. For example, we may construct A from its rows as follows:

  arow1 = [ 2 4 ];
  arow2 = [ 6 8 ];
  arow3 = [ 10 12 ];
  A = [ arow1; arow2; arow3 ];

Similarly, we can build a matrix from its columns:

  acol1 = [2 6 10]';
  acol2 = [4; 8; 12];
  A = [ acol1 acol2 ];

Using the ":" operator to create arrays

Sometimes we have vectors that have specific patterns. For example, a=[1 2 3 4 5]. The : operator (see here for MATLAB's explanation of this) can help us create such arrays:

 array = lower : increment : upper;

This creates an array starting at lower and ending at upper with spacing of increment. You may eliminate the increment argument, in which case it defaults to 1.

Examples:

  a = 1:5;        % creates a=[ 1 2 3 4 5 ]
  b = 1:1:5;      % creates b=[ 1 2 3 4 5 ]
  c = 1:2:7;      % creates c=[ 1 3 5 7 ]
  d = 1:2:8;      % creates d=[ 1 3 5 7 ]
  e = 1.1:5       % creates e=[ 1.1 2.1 3.1 4.1 ]
  f = 0.1:0.1:0.5 % creates f=[ 0.1 0.2 0.3 0.4 0.5 ]
  g = 5:-1:1;     % creates g=[ 5 4 3 2 1 ];
  h = (1:3)';     % creates a column vector with elements 1, 2, 3.

Linspace and Logspace

Occasionally we want to create numbers that are equally spaced between two limits. We can use the linspace command to do this.

linspace( lo, hi, n );
Creates a row vector with n points equally spaced between lo and hi.
The last argument, n is not required. If omitted, 100 points will be used.

Examples:

  a = linspace(1,5,5);    % creates a=[ 1 2 3 4 5 ]
  b = linspace(1,5,4);    % creates b=[ 1.0 2.333 3.667 5.0 ]
  c = linspace(5,1,5);    % creates c=[ 5 4 3 2 1 ]
  d = linspace(1,5,5)';   % creates a column vector
  e = linspace(1,5);      % uses 100 points between 1 and 5.
  lo=0; hi=10;
  f = linspace(lo,hi);    % uses 100 points between lo (0) and hi (10)

Sometimes we want points logarithmically spaced. For example, we may want to create a vector a=[ 0.1 1 10 100]. This can be accomplished using the logspace command:

logspace( lo, hi, n );
Creates a vector with n points spaced logarithmically between \(10^\mathrm{lo}\) and \(10^\mathrm{hi}\)
The last argument, n is not required. If omitted, 50 points will be used.

Examples:

  a = logspace(-1,3,5);    % creates a=[ 0.1 1 10 100 1000 ]
  b = logspace(3,-1,5);    % creates b=[ 1000 100 10 1 0.1 ]
  c = logspace(-1,3,5)';   % creates a column vector
  d = logspace(-3,3);      % 50 points between 10^-3 and 10^3.

Shortcuts for Special Matrices

ones(nrow,ncol)
creates an array of all ones with nrow rows and ncol columns. [1]
zeros(nrow,ncol)
creates an array of all zeros with nrow rows and ncol columns. [2]
eye(nrow,ncol)
creates an array with ones on its diagonal and zeros elsewhere. [3]
Examples of creating matrices.
Matlab code Resulting array
eye(3,2)
\(\left[\begin{smallmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{smallmatrix}\right]\)
eye(3,3)
\(\left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{smallmatrix} \right] \)
rand(nrow,ncol)
creates an array with nrow rows and ncol columns full of random numbers. [4]

For more information on creating matrices, type help elmat in the command window.

Sparse Matrices

Error creating thumbnail: Unable to save thumbnail to destination
This section is a stub and needs to be expanded.
If you can provide information or finish this section you're welcome to do so and then remove this message afterwards.


Array Arithmetic

One of the most powerful features of MATLAB is the ability to perform operations on entire arrays at once.

Addition and Subtraction

Given two arrays of the same size, one may add or subtract these as follows:

  C = A + B;
Example 1:
  A = rand(3);    % A is 3x3 with random entries
  B = ones(3);    % B is 3x3 with 1 everywhere
  C = A+B;        % C is 3x3 formed by adding corresponding elements of A & B
  D = B + eye(3); % Add 1 to each diagonal element in B
Example 2:
  a = [1 2 3];   % 1x3 matrix (row vector)
  b = [4 5 6];   % 1x3 matrix (row vector)
  c = a+b;       % add a and b
  d = a' + b';   % d is a columb vector
Example 3:
  A = [1 2 3; 4 5 6];  % A is 2x3
  a = [-1 -2 -3];      % a is 1x3
  b = a + A(1,:);      % add first row of A to a

Multiplication

Multiplication of two arrays, C=A*B requires that the number of columns in A is equal to the number of rows in B. The array C has the same number of rows as A and same number of columns as B.

Example 1
  A = rand(3);
  B = rand(3);
  C = A*B;
A, B and C are all 3x3 matrices.
Example 2
  A = rand(3);
  b = rand(3,1);
  c = A*B;
A is a 3x3 matrix, b and c are column vectors
Example 3
  A = rand(3,5);
  B = rand(5,2);
  C = A*B;
A is 3x5, B is 5x2 and C is 3x2


Elemental Operations

The following table summarizes the elemental operators available in MATLAB. Each of these operators is discussed in more detail below.

Summary of Elemental Operations
Operator Description
.* Elemental Multiplication
./ Elemental Division
.^ Elemental Exponentiation

Elemental Multiplication

Arrays of the same size may be multiplied or divided element-by-element using the .* operator so that C = A .* B produces an array whose elements are given by \(C_{i,j} = A_{i,j} B_{i,j}\).


Matlab Code Resulting Arrays Comments
  A = [1 2; 3 4];
  B = [0 1; 5 2];
  C = A .* B
\(A = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]\),

\(B = \left[\begin{smallmatrix} 0 & 1 \\ 5 & 2 \end{smallmatrix}\right]\), \(C = \left[\begin{smallmatrix} 0 & 2 \\ 15 & 8 \end{smallmatrix}\right]\)

C is obtained by multiplying corresponding entries in A and B.
  a = [1 2 3];
  b = [2; 4; 6];
  c = a .* b';
\(a = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\),

\(b = \begin{bmatrix} 2 & 4 & 6 \end{bmatrix}\), \(c = \begin{bmatrix} 2 & 8 & 18 \end{bmatrix}\)

Note that elemental multiplication is very different from array multiplication. For example,
  d = a*b;
  e = b*a;

would produce \(d = 24\), and \( e = \left[\begin{smallmatrix} 2 & 4 & 6 \\ 4 & 8 & 12 \\ 6 & 12 & 18 \end{smallmatrix}\right] \).

If you need to brush up on this, see the tutorial on matrix multiplication.

Elemental Division

The . / operator performs elemental division on two arrays of the same size.

Matlab Code Results
  a = [ 2 4 6 ];
  b = [ 1 2 3 ];
  c = a./b;
  d = b./a;
c=[2 2 2] and d=[0.5 0.5 0.5].
  A = [1 2 6; 12 5 6];
  B = [2 2 -3; 6 1 -2];
  C = A./B;
  D = B./A;
\(C=\left[\begin{smallmatrix} 0.5 & 1 & -2 \\ 2 & 5 & -3 \end{smallmatrix}\right]\)

\(D=\left[\begin{smallmatrix} 2 & 1 & -0.5 \\ 0.5 & 0.2 & -0.3333 \end{smallmatrix}\right]\)

Elemental Exponentiation

A .^ b
results in each element in A being raised to the power b. Here b must be a scalar, not a matrix or a vector.
Matlab Code Results
  x = linspace(-pi,pi);
  y = sin(x);
  z = cos(x);
  w = y.^2 + z.^2;
All arrays are row vectors with 100 elements since x was formed this way using the linspace command. The w array has all entries equal to 1.

NOTE: Elemental exponentiation is very different from array exponentiation. For example:

  A = [1 2; 3 4];
  B = A.^3;
  C = A^3;

results in \(B = \left[\begin{smallmatrix} 1 & 8 \\ 27 & 64 \end{smallmatrix}\right]\) and \(C = \left[\begin{smallmatrix} 37 & 54 \\ 81 & 118 \end{smallmatrix} \right] \). Here we have C=A*A*A, where these are matrix-matrix multiplications.

Accessing Arrays

Often we need to access certain elements or groups of elements from an array. There are a few ways to do this.

Indexing arrays

Recall that an array is simply a collection of numbers. Arrays may be 1-dimensional (vectors), 2-dimensional (matrices) or even higher dimensional. Each number in an array has a unique location identified by an index. Array indexing is best illustrated by example.

  a = 2:2:10;  % creates a row vector: [2 4 6 8 10]
  a1 = a(1);   % assigns the first element of a to "a1"

We can likewise index into a matrix as:

  B = [ 5 2 9; 1 7 3; 2 6 0];
  var = B(2,3);      % var = 3  (row 2, column 3)

In general for a two-dimensional array, B(i,j) refers to the ith row and jth column of B. This is extensible to arrays of higher dimensions as well.

If we return to our previous example with the B matrix, we find something interesting:

  B = [ 5 2 9; 1 7 3; 2 6 0];
  b = B(5);

What should the value of b be? One might expect this to result in an error, since B is not a vector. However, this returns b=7. What is happening? Matlab actually treats a matrix as a long vector, B = [ 5 2 9 1 7 3 2 6 0 ], with each row stacked next to the previous row. Therefore, B(5) says "go to the fifth entry" counted along rows.


Slicing arrays using the ":" operator

Sometimes we need to grab a group of elements from an array. We can do this using the : operator For example:

  A = [ 1 2 3; 4 5 6; 7 8 9 ];
  col1 = A(:,1);  % all rows, first column
  row2 = A(2,:);  % second row, all columns

Here : acts to select all elements in the row or column of A. We can grab a subset of rows or columns by the following:

  A = [ 1 2 3; 4 5 6; 7 8 9 ];
  a = A( 1:2, 2:3 );

This takes rows 1 and 2 and columns 2 and 3 in A and creates a new array a. Now a looks like \(a=\begin{bmatrix}2 & 3 \\ 5 & 6 \end{bmatrix}\).

The : operator is a very powerful tool to extract portions of an array.


Information about Arrays

size
returns the number of rows and columns in an array. For example,
A = [1 2 3; 4 5 6];
[nrow,ncol] = size(A);
The result of this is that nrow=2 and ncol=3.
length
returns the number of elements in a vector. Example:
a = [1 2 3 4 3 2 1];
na = length(a);
Results in na=7.
max
For a vector, max(v) returns the largest element in v. For matrices, max(V) returns a row vector whose elements contain the maximum value in each column of V.
min
Analogous to max.


Other Useful Tools

Use MATLAB's help command to learn more about how to use these commands.

Matlab command Summary Example(s)
sum
  • For a vector, sum(v) returns the sum of the elements of v.
  • For matrices, sum(V) returns a row vector whose elements contain the sum of the columns of V.
a=sum([1 2 3]);    % a=6
b=sum([1 2; 3 4]); % b = [4 6]
sort Sort an array in ascending or descending order.
sort([2 5 1])  %  gives '''1 2 5'''
any any(A) Evaluates to "true" if any of the elements in the array are true (nonzero).
a=[1 2 3]; b=[1 3 4];
r1 = any(a>b);    % r1 is FALSE (equal to 0)
r2 = any(a==b);   % r2 is TRUE (equal to 1) since a(1) is equal to b(1).
all Evaluates to "true" if all of the elements in the array are true (nonzero).
A = [1 2; 3 4];
B = [-1 1; 2 3];
r1 = all(A>B);   % r1 is TRUE (equal to 1) since the elements in A are
                 % all greater than their corresponding elements in B.
r2 = all(2*B>A); % r2 is FALSE
find find(A) returns the indices where A is nonzero.
A = [1 5 0; 0 2 0];
i = find(A(:,1));  % i=1
i = find(A(:,2));  % i=[1 2]'
i = find(A);       % i=[1 3 4]'
[i,j] = find(A);   % i=[ 1 1 2 ]'  j=[ 1 2 2 ]'
i = find(A>4);     % i=3
[i,j] = find(A<2); % i=[1 2 1 2]'  j=[1 1 3 3]'
isequal isequal(A,B) returns true if all elements of A and B are equal, false otherwise.
A = [1 2; 3 4];
B = [2 4; 6 8];
c = isequal(A,B);   % c is FALSE
d = isequal(2*A,B); % d is TRUE

Here is something a bit more complex, using the fact that \(\sin^2(\alpha)+\cos^2(\alpha) = 1\)

x = rand(5);
y1 = sin(x).*sin(x);
z1 = cos(x).*cos(x);
q = ones(5);
c = isequal(q,y1+z1);  % c is TRUE


Cell Arrays

The arrays we have discussed up to this point can hold a single type of information. For example, each element of the array can hold a single number. They could also hold a text string, as long as each text string was the same length.

Cell arrays are special kinds of arrays that can hold any type of information. For example:

  a = { 1, 'hi there', [5 4 3] };

Cell arrays are built in much the same way as a normal array with one important difference: we use { and } rather than [ and ] to construct the array. Accessing cell arrays is also different. For example, given the cell array a defined above, we could access its elements in the following way:

  a{1}   % refers to 1
  a{2}   % refers to 'hi there'
  a{3}   % refers to the array [5 4 3]

Cell arrays are particularly convenient when working with text strings. For example, if we wanted to create a list of chemical compounds along with their molecular weights, we could do the following:

names = { 'Carbon Dioxide (CO2)','Water (H2O)','Methane (CH4)' };
mw = [ 44, 18, 16 ];
fprintf('\n  Compound        Molecular Weight\n');
fprintf('    Name               (g/mol)\n');
fprintf('-----------------------------------\n');
for i=1:length(mw)
  fprintf('%20s   %3.1f\n',names{i},mw(i));
end
fprintf('-----------------------------------\n');
Note: follow these links for help on for loops and fprintf