Octave:Vectors and matrices

From AIMSWiki

Table of contents
1 Creating vectors and matrices
2 Operators

2.1 Useful Functions
2.2 Element operations

3 Indexing

3.1 Ranges

4 Functions

4.1 Creating matrices

4.1.1 Other matrices

4.2 Changing matrices

5 Linear algebra


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Creating vectors and matrices

Here is how we specify a row vector in Octave: 

octave:1> x = [1, 3, 2]
x =

  1  3  2

Note that

  • the vector is enclosed in square brackets;
  • each entry is separated by a comma

To specify a column vector, we simply replace the commas with semicolons: 

octave:2> x = [1; 3; 2]
x =

  1
  3
  2

From this you can see that we use a comma to go to the next column of a vector (or matrix) and a semicolon to go to the next row. So, to specify a matrix, type in the rows (separating each entry with a comma) and use a semicolon to go to the next row. 

octave:3> A = [1, 1, 2; 3, 5, 8; 13, 21, 34]
A =

   1   1   2
   3   5   8
  13  21  34
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Operators

You can use the standard operators to

  • add (+),
  • subtract (-), and
  • multiply (*)

matrices, vectors and scalars with one another. Note that the matrices need to have matching dimensions (inner dimensions in the case of multiplication) for these operators to work.

  • The transpose operator is the single quote: '. To continue from the example in the previous section, 
octave:4> A'
ans =

   1   3  13
   1   5  21
   2   8  34

(Note: this is actually the complex conjugate transpose operator, but for real matrices this is the same as the transpose. To compute the transpose of a complex matrix, use the dot transpose (.') operator.)

  • The power operator (^) can also be used to compute real powers of square matrices.
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Useful Functions

  • sum()
  • mean()
  • max(), min()
  • size(), length()
  • find()
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Element operations

When you have two matrices of the same size, you can perform element by element operations on them. For example, the following divides each element of A by the corresponding element in B: 

octave:1> A = [1, 6, 3; 2, 7, 4]
A =

  1  6  3
  2  7  4

octave:2> B = [2, 7, 2; 7, 3, 9]
B =

  2  7  2
  7  3  9

octave:3> A ./ B
ans =

  0.50000  0.85714  1.50000
  0.28571  2.33333  0.44444

Note that you use the dot divide (./) operator to perform element by element division. There are similar operators for multiplication (.*) and exponentiation (.^).

The dot divide operators can also be used together with scalars in the following manner. 

C = a ./ B

returns a matrix, C where each entry is defined by

C_{ij} = \frac{a}{B_{ij}},

i.e. a is divided by each entry in B. Similarly 

C = a .^ B

return a matrix with

C_{ij} = a^{B_{ij}}.
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Indexing

You can work with parts of matrices and vectors by indexing into them. You use a vector of integers to tell Octave which elements of a vector or matrix to use. For example, we create a vector 

octave:1> x = [1.2, 5, 7.6, 3, 8]
x =

  1.2000  5.0000  7.6000  3.0000  8.0000

Now, to see the second element of x, type 

octave:2> x(2)
ans = 5

You can also view a list of elements as follows. 

octave:3> x([1, 3, 4])
ans =

  1.2000  7.6000  3.0000

This last command displays the 1st, 3rd and 4th elements of the vector x.

To select rows and columns from a matrix, we use the same principle. Let's define a matrix 

octave:4> A = [1, 2, 3; 4, 5, 6; 7, 8, 9]
A =

  1  2  3
  4  5  6
  7  8  9

and select the 1st and 3rd rows and 2nd and 3rd columns: 

octave:5> A([1, 3], [2, 3])
ans =
      2  3
      8  9

The colon operator (:) can be used to select all rows or columns from a matrix. So, to select all the elements from the 2nd row, type 

octave:6> A(2, :)
ans =

  4  5  6
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Ranges

We can also select a range of rows or columns from a matrix. We specify a range with 

start:step:stop

The first number in the range is start, the second is start+step, the third, start+2*step, etc. The last number is less than or equal to stop.

You can actually type ranges at the Octave prompt to see what the results are. For example, 

octave:3> 1:3:10
ans =

   1   4   7  10

Often, you simply want the step size to be 1. In this case, you can leave out the step parameter and type octave:4> 1:10 

ans =

   1   2   3   4   5   6   7   8   9  10

As you can see, the result of a range command is simply a vector of integers. We can now use this to index into a vector or matrix. To select the 2\times 2 submatrix at the top left of A, use 

octave:4> A(1:2, 1:2)
ans =

  1  2
  4  5

Finally, there is a keyword called end that can be used when indexing into a matrix or vector. It refers to the last element in the row or column. To index the last 3 elements of the vector x, use 

octave:5> x(end-2:end)
ans =

  7.6000  3.0000  8.0000
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Functions

The following functions can be used to create and manipulate matrices.

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Creating matrices

  • tril(A) returns the lower triangular part of A.
  • triu(A) returns the upper triangular part of A.
  • eye(n) returns the n\times n identity matrix. You can also use eye(m, n) to return m\times n rectangular identity matrices.
  • ones(m, n) returns an m\times n matrix filled with 1s.
  • zeros(m, n) returns an m\times n matrix filled with 0s.
  • rand(m, n) returns an m\times n matrix filled with random elements drawn uniformly from [0,1).
  • randn(m, n) returns an m\times n matrix filled with normally distributed random elements.
  • randperm(n) returns a row vector containing a random permutation of the numbers 1, 2, \ldots, n.
  • diag(x) or diag(A). For a vector, x, this returns a square matrix with the elements of x on the diagonal and 0s everywhere else. For a matrix, A, this returns a vector containing the diagonal elements of A. For example, 
octave:16> A = [1, 2, 3; 4, 5, 6; 7, 8, 9]
A =

  1  2  3
  4  5  6
  7  8  9

octave:17> x = diag(A)
ans =

  1
  5
  9

octave:18> diag(x)
ans =

  1  0  0
  0  5  0
  0  0  9
  • linspace(a, b, n) returns a vector with n values, such that the first element equals a, the last element equals b and the difference between consecutive elements is constant. The last argument, n, is optional with default value 100.
  • logspace(a, b, n) returns a vector with n values, such that the first element equals 10a, the last element equals 10b and the ratio between consecutive elements is constant. The last argument, n is optional with default value 50.
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Other matrices

There are some more functions for creating special matrices. These are

  • hankel (Hankel matrix),
  • hilb (Hilbert matrix),
  • invhilb (Inverse of a Hilbert matrix),
  • sylvester_matrix (Sylvester matrix),
  • toeplitz (Toeplitz matrix),
  • vander (Vandermonde matrix).

Use help to find out more about how to use these functions.

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Changing matrices

  • fliplr(A) returns a copy of matrix A with the order of the columns reversed, for example, 
octave:49> A = [1, 2, 3, 4; 5, 6, 7, 8; 9, 10, 11, 12] 
A =

   1   2   3   4
   5   6   7   8
   9  10  11  12

octave:50> fliplr(A)
ans =

   4   3   2   1
   8   7   6   5
  12  11  10   9
  • flipud(A) returns a copy of matrix A with the order of the rows reversed, for example, 
octave:51> flipud(A)
ans =

   9  10  11  12
   5   6   7   8
   1   2   3   4
  • rot90(A, n) returns a copy of matrix A that has been rotated by (90n)° counterclockwise. The second argument, n, is optional with default value 1. 
octave:52> rot90(A)
ans =

   4   8  12
   3   7  11
   2   6  10
   1   5   9
  • reshape(A, m, n) creates an m\times n matrix with elements taken from A. The number of elements in A has to be equal to mn. The elements are taken from A in column major order, meaning that values in the first column (A_{11}, \ldots, A_{m1}) are read first, then the second column (A_{12}, \ldots, A_{m2}), etc. 
octave:53> reshape(A, 2, 6)
ans =

   1   9   6   3  11   8
   5   2  10   7   4  12
  • sort(x) returns a copy of the vector x with the elements sorted in increasing order. 
octave:54> x = rand(1, 6)
x =

  0.25500  0.33525  0.26586  0.92658  0.68799  0.69682

octave:55> sort(x)
ans =

  0.25500  0.26586  0.33525  0.68799  0.69682  0.92658
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Linear algebra

For a description of more operators and functions that can be used to manipulate vectors and matrices, find eigenvalues, etc., see the Linear algebra section.

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