# 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 (`"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 (`"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.

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