Special Matrices

Octave provides a number of functions for creating special matrix forms. In nearly all cases, it is best to use the built-in functions for this purpose than to try to use other tricks to achieve the same effect.

Special Utility Matrices

Built-in Function: eye (x)

Built-in Function: eye (n, m)

Returns an identity matrix. If invoked with a single scalar argument, eye returns a square matrix with the dimension specified. If you supply two scalar arguments, eye takes them to be the number of rows and columns. If given a vector with two elements, eye uses the values of the elements as the number of rows and columns, respecively. For example,

eye (3)

     =>  1  0  0
         0  1  0
         0  0  1

The following expressions all produce the same result:

eye (2)
eye (2, 2)
eye (size ([1, 2; 3, 4])

For compatibility with MATLAB, calling eye with no arguments is equivalent to calling it with an argument of 1.

Built-in Function: ones (x)

Built-in Function: ones (n, m)

Returns a matrix whose elements are all 1. The arguments are handled the same as the arguments for eye.

If you need to create a matrix whose values are all the same, you should use an expression like

val_matrix = val * ones (n, m)

Built-in Function: zeros (x)

Built-in Function: zeros (n, m)

Returns a matrix whose elements are all 0. The arguments are handled the same as the arguments for eye.

Loadable Function: rand (x)

Loadable Function: rand (n, m)

Loadable Function: rand ("seed", x)

Returns a matrix with random elements uniformly distributed on the interval (0, 1). The arguments are handled the same as the arguments for eye. In addition, you can set the seed for the random number generator using the form

randn ("seed", x)

where x is a scalar value. If called as

rand ("seed")

rand returns the current value of the seed.

Loadable Function: randn (x)

Loadable Function: randn (n, m)

Loadable Function: randn ("seed", x)

Returns a matrix with normally distributed random elements. The arguments are handled the same as the arguments for eye. In addition, you can set the seed for the random number generator using the form

randn ("seed", x)

where x is a scalar value. If called as

randn ("seed")

randn returns the current value of the seed.

The rand and randn functions use separate generators. This ensures that

rand ("seed", 13);
randn ("seed", 13);
u = rand (100, 1);
n = randn (100, 1);

and

rand ("seed", 13);
randn ("seed", 13);
u = zeros (100, 1);
n = zeros (100, 1);
for i = 1:100
  u(i) = rand ();
  n(i) = randn ();
end

produce equivalent results.

Normally, rand and randn obtain their initial seeds from the system clock, so that the sequence of random numbers is not the same each time you run Octave. If you really do need for to reproduce a sequence of numbers exactly, you can set the seed to a specific value.

If it is invoked without arguments, rand and randn return a single element of a random sequence.

The rand and randn functions use Fortran code from RANLIB, a library of fortran routines for random number generation, compiled by Barry W. Brown and James Lovato of the Department of Biomathematics at The University of Texas, M.D. Anderson Cancer Center, Houston, TX 77030.

Built-in Function: diag (v, k)

Returns a diagonal matrix with vector v on diagonal k. The second argument is optional. If it is positive, the vector is placed on the k-th super-diagonal. If it is negative, it is placed on the -k-th sub-diagonal. The default value of k is 0, and the vector is placed on the main diagonal. For example,

diag ([1, 2, 3], 1)

     =>  0  1  0  0
         0  0  2  0
         0  0  0  3
         0  0  0  0

The functions linspace and logspace make it very easy to create vectors with evenly or logarithmically spaced elements. See section Ranges.

Function File: linspace (base, limit, n)

creates a row vector with n (n greater than 1) linearly spaced elements between base and limit. The base and limit are always included in the range. If base is greater than limit, the elements are stored in decreasing order. If the number of points is not specified, a value of 100 is used.

The linspace function always returns a row vector, regardless of the value of prefer_column_vectors.

Function File: logspace (base, limit, n)

Similar to linspace except that the values are logarithmically spaced from

If limit is equal to the points are between not in order to be compatible with the corresponding MATLAB function.

Famous Matrices

The following functions return famous matrix forms.

Function File: hadamard (k)

Return the Hadamard matrix of order n = 2^k.

Function File: hankel (c, r)

Return the Hankel matrix constructed given the first column c, and (optionally) the last row r. If the last element of c is not the same as the first element of r, the last element of c is used. If the second argument is omitted, the last row is taken to be the same as the first column.

A Hankel matrix formed from an m-vector c, and an n-vector r, has the elements

Function File: hilb (n)

Return the Hilbert matrix of order n. The element of a Hilbert matrix is defined as

Function File: invhilb (n)

Return the inverse of a Hilbert matrix of order n. This is exact. Compare with the numerical calculation of inverse (hilb (n)), which suffers from the ill-conditioning of the Hilbert matrix, and the finite precision of your computer's floating point arithmetic.

Function File: toeplitz (c, r)

Return the Toeplitz matrix constructed given the first column c, and (optionally) the first row r. If the first element of c is not the same as the first element of r, the first element of c is used. If the second argument is omitted, the first row is taken to be the same as the first column.

A square Toeplitz matrix has the form

Function File: vander (c)

Return the Vandermonde matrix whose next to last column is c.

A Vandermonde matrix has the form

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