# `linalg.special_matrices`¶

## Module Contents¶

### Functions¶

 `tri`(N,M=None,k=0,dtype=None) Construct (N, M) matrix filled with ones at and below the k-th diagonal. `tril`(m,k=0) Make a copy of a matrix with elements above the k-th diagonal zeroed. `triu`(m,k=0) Make a copy of a matrix with elements below the k-th diagonal zeroed. `toeplitz`(c,r=None) Construct a Toeplitz matrix. `circulant`(c) Construct a circulant matrix. `hankel`(c,r=None) Construct a Hankel matrix. `hadamard`(n,dtype=int) Construct a Hadamard matrix. `leslie`(f,s) Create a Leslie matrix. `kron`(a,b) Kronecker product. `block_diag`(*arrs) Create a block diagonal matrix from provided arrays. `companion`(a) Create a companion matrix. `helmert`(n,full=False) Create a Helmert matrix of order n. `hilbert`(n) Create a Hilbert matrix of order n. `invhilbert`(n,exact=False) Compute the inverse of the Hilbert matrix of order n. `pascal`(n,kind=”symmetric”,exact=True) Returns the n x n Pascal matrix. `invpascal`(n,kind=”symmetric”,exact=True) Returns the inverse of the n x n Pascal matrix. `dft`(n,scale=None) Discrete Fourier transform matrix.
`tri`(N, M=None, k=0, dtype=None)

Construct (N, M) matrix filled with ones at and below the k-th diagonal.

The matrix has A[i,j] == 1 for i <= j + k

N : int
The size of the first dimension of the matrix.
M : int or None, optional
The size of the second dimension of the matrix. If M is None, M = N is assumed.
k : int, optional
Number of subdiagonal below which matrix is filled with ones. k = 0 is the main diagonal, k < 0 subdiagonal and k > 0 superdiagonal.
dtype : dtype, optional
Data type of the matrix.
tri : (N, M) ndarray
Tri matrix.
```>>> from scipy.linalg import tri
>>> tri(3, 5, 2, dtype=int)
array([[1, 1, 1, 0, 0],
[1, 1, 1, 1, 0],
[1, 1, 1, 1, 1]])
>>> tri(3, 5, -1, dtype=int)
array([[0, 0, 0, 0, 0],
[1, 0, 0, 0, 0],
[1, 1, 0, 0, 0]])
```
`tril`(m, k=0)

Make a copy of a matrix with elements above the k-th diagonal zeroed.

m : array_like
Matrix whose elements to return
k : int, optional
Diagonal above which to zero elements. k == 0 is the main diagonal, k < 0 subdiagonal and k > 0 superdiagonal.
tril : ndarray
Return is the same shape and type as m.
```>>> from scipy.linalg import tril
>>> tril([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 0,  0,  0],
[ 4,  0,  0],
[ 7,  8,  0],
[10, 11, 12]])
```
`triu`(m, k=0)

Make a copy of a matrix with elements below the k-th diagonal zeroed.

m : array_like
Matrix whose elements to return
k : int, optional
Diagonal below which to zero elements. k == 0 is the main diagonal, k < 0 subdiagonal and k > 0 superdiagonal.
triu : ndarray
Return matrix with zeroed elements below the k-th diagonal and has same shape and type as m.
```>>> from scipy.linalg import triu
>>> triu([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 1,  2,  3],
[ 4,  5,  6],
[ 0,  8,  9],
[ 0,  0, 12]])
```
`toeplitz`(c, r=None)

Construct a Toeplitz matrix.

The Toeplitz matrix has constant diagonals, with c as its first column and r as its first row. If r is not given, `r == conjugate(c)` is assumed.

c : array_like
First column of the matrix. Whatever the actual shape of c, it will be converted to a 1-D array.
r : array_like, optional
First row of the matrix. If None, `r = conjugate(c)` is assumed; in this case, if c[0] is real, the result is a Hermitian matrix. r[0] is ignored; the first row of the returned matrix is `[c[0], r[1:]]`. Whatever the actual shape of r, it will be converted to a 1-D array.
A : (len(c), len(r)) ndarray
The Toeplitz matrix. Dtype is the same as `(c[0] + r[0]).dtype`.

circulant : circulant matrix hankel : Hankel matrix solve_toeplitz : Solve a Toeplitz system.

The behavior when c or r is a scalar, or when c is complex and r is None, was changed in version 0.8.0. The behavior in previous versions was undocumented and is no longer supported.

```>>> from scipy.linalg import toeplitz
>>> toeplitz([1,2,3], [1,4,5,6])
array([[1, 4, 5, 6],
[2, 1, 4, 5],
[3, 2, 1, 4]])
>>> toeplitz([1.0, 2+3j, 4-1j])
array([[ 1.+0.j,  2.-3.j,  4.+1.j],
[ 2.+3.j,  1.+0.j,  2.-3.j],
[ 4.-1.j,  2.+3.j,  1.+0.j]])
```
`circulant`(c)

Construct a circulant matrix.

c : (N,) array_like
1-D array, the first column of the matrix.
A : (N, N) ndarray
A circulant matrix whose first column is c.

toeplitz : Toeplitz matrix hankel : Hankel matrix solve_circulant : Solve a circulant system.

New in version 0.8.0.

```>>> from scipy.linalg import circulant
>>> circulant([1, 2, 3])
array([[1, 3, 2],
[2, 1, 3],
[3, 2, 1]])
```
`hankel`(c, r=None)

Construct a Hankel matrix.

The Hankel matrix has constant anti-diagonals, with c as its first column and r as its last row. If r is not given, then r = zeros_like(c) is assumed.

c : array_like
First column of the matrix. Whatever the actual shape of c, it will be converted to a 1-D array.
r : array_like, optional
Last row of the matrix. If None, `r = zeros_like(c)` is assumed. r[0] is ignored; the last row of the returned matrix is `[c[-1], r[1:]]`. Whatever the actual shape of r, it will be converted to a 1-D array.
A : (len(c), len(r)) ndarray
The Hankel matrix. Dtype is the same as `(c[0] + r[0]).dtype`.

toeplitz : Toeplitz matrix circulant : circulant matrix

```>>> from scipy.linalg import hankel
>>> hankel([1, 17, 99])
array([[ 1, 17, 99],
[17, 99,  0],
[99,  0,  0]])
>>> hankel([1,2,3,4], [4,7,7,8,9])
array([[1, 2, 3, 4, 7],
[2, 3, 4, 7, 7],
[3, 4, 7, 7, 8],
[4, 7, 7, 8, 9]])
```
`hadamard`(n, dtype=int)

Constructs an n-by-n Hadamard matrix, using Sylvester’s construction. n must be a power of 2.

n : int
The order of the matrix. n must be a power of 2.
dtype : dtype, optional
The data type of the array to be constructed.
H : (n, n) ndarray

New in version 0.8.0.

```>>> from scipy.linalg import hadamard
array([[ 1.+0.j,  1.+0.j],
[ 1.+0.j, -1.-0.j]])
array([[ 1,  1,  1,  1],
[ 1, -1,  1, -1],
[ 1,  1, -1, -1],
[ 1, -1, -1,  1]])
```
`leslie`(f, s)

Create a Leslie matrix.

Given the length n array of fecundity coefficients f and the length n-1 array of survival coefficents s, return the associated Leslie matrix.

f : (N,) array_like
The “fecundity” coefficients.
s : (N-1,) array_like
The “survival” coefficients, has to be 1-D. The length of s must be one less than the length of f, and it must be at least 1.
L : (N, N) ndarray
The array is zero except for the first row, which is f, and the first sub-diagonal, which is s. The data-type of the array will be the data-type of `f[0]+s[0]`.

New in version 0.8.0.

The Leslie matrix is used to model discrete-time, age-structured population growth [1]_ [2]_. In a population with n age classes, two sets of parameters define a Leslie matrix: the n “fecundity coefficients”, which give the number of offspring per-capita produced by each age class, and the n - 1 “survival coefficients”, which give the per-capita survival rate of each age class.

 [1] P. H. Leslie, On the use of matrices in certain population mathematics, Biometrika, Vol. 33, No. 3, 183–212 (Nov. 1945)
 [2] P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, Vol. 35, No. 3/4, 213–245 (Dec. 1948)
```>>> from scipy.linalg import leslie
>>> leslie([0.1, 2.0, 1.0, 0.1], [0.2, 0.8, 0.7])
array([[ 0.1,  2. ,  1. ,  0.1],
[ 0.2,  0. ,  0. ,  0. ],
[ 0. ,  0.8,  0. ,  0. ],
[ 0. ,  0. ,  0.7,  0. ]])
```
`kron`(a, b)

Kronecker product.

The result is the block matrix:

```a[0,0]*b    a[0,1]*b  ... a[0,-1]*b
a[1,0]*b    a[1,1]*b  ... a[1,-1]*b
...
a[-1,0]*b   a[-1,1]*b ... a[-1,-1]*b
```
a : (M, N) ndarray
Input array
b : (P, Q) ndarray
Input array
A : (M*P, N*Q) ndarray
Kronecker product of a and b.
```>>> from numpy import array
>>> from scipy.linalg import kron
>>> kron(array([[1,2],[3,4]]), array([[1,1,1]]))
array([[1, 1, 1, 2, 2, 2],
[3, 3, 3, 4, 4, 4]])
```
`block_diag`(*arrs)

Create a block diagonal matrix from provided arrays.

Given the inputs A, B and C, the output will have these arrays arranged on the diagonal:

```[[A, 0, 0],
[0, B, 0],
[0, 0, C]]
```
A, B, C, … : array_like, up to 2-D
Input arrays. A 1-D array or array_like sequence of length n is treated as a 2-D array with shape `(1,n)`.
D : ndarray
Array with A, B, C, … on the diagonal. D has the same dtype as A.

If all the input arrays are square, the output is known as a block diagonal matrix.

Empty sequences (i.e., array-likes of zero size) will not be ignored. Noteworthy, both [] and [[]] are treated as matrices with shape `(1,0)`.

```>>> from scipy.linalg import block_diag
>>> A = [[1, 0],
...      [0, 1]]
>>> B = [[3, 4, 5],
...      [6, 7, 8]]
>>> C = [[7]]
>>> P = np.zeros((2, 0), dtype='int32')
>>> block_diag(A, B, C)
array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 3, 4, 5, 0],
[0, 0, 6, 7, 8, 0],
[0, 0, 0, 0, 0, 7]])
>>> block_diag(A, P, B, C)
array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 3, 4, 5, 0],
[0, 0, 6, 7, 8, 0],
[0, 0, 0, 0, 0, 7]])
>>> block_diag(1.0, [2, 3], [[4, 5], [6, 7]])
array([[ 1.,  0.,  0.,  0.,  0.],
[ 0.,  2.,  3.,  0.,  0.],
[ 0.,  0.,  0.,  4.,  5.],
[ 0.,  0.,  0.,  6.,  7.]])
```
`companion`(a)

Create a companion matrix.

Create the companion matrix [1]_ associated with the polynomial whose coefficients are given in a.

a : (N,) array_like
1-D array of polynomial coefficients. The length of a must be at least two, and `a[0]` must not be zero.
c : (N-1, N-1) ndarray
The first row of c is `-a[1:]/a[0]`, and the first sub-diagonal is all ones. The data-type of the array is the same as the data-type of `1.0*a[0]`.
ValueError
If any of the following are true: a) `a.ndim != 1`; b) `a.size < 2`; c) `a[0] == 0`.

New in version 0.8.0.

 [1] R. A. Horn & C. R. Johnson, Matrix Analysis. Cambridge, UK: Cambridge University Press, 1999, pp. 146-7.
```>>> from scipy.linalg import companion
>>> companion([1, -10, 31, -30])
array([[ 10., -31.,  30.],
[  1.,   0.,   0.],
[  0.,   1.,   0.]])
```
`helmert`(n, full=False)

Create a Helmert matrix of order n.

This has applications in statistics, compositional or simplicial analysis, and in Aitchison geometry.

n : int
The size of the array to create.
full : bool, optional
If True the (n, n) ndarray will be returned. Otherwise the submatrix that does not include the first row will be returned. Default: False.
M : ndarray
The Helmert matrix. The shape is (n, n) or (n-1, n) depending on the full argument.
```>>> from scipy.linalg import helmert
>>> helmert(5, full=True)
array([[ 0.4472136 ,  0.4472136 ,  0.4472136 ,  0.4472136 ,  0.4472136 ],
[ 0.70710678, -0.70710678,  0.        ,  0.        ,  0.        ],
[ 0.40824829,  0.40824829, -0.81649658,  0.        ,  0.        ],
[ 0.28867513,  0.28867513,  0.28867513, -0.8660254 ,  0.        ],
[ 0.2236068 ,  0.2236068 ,  0.2236068 ,  0.2236068 , -0.89442719]])
```
`hilbert`(n)

Create a Hilbert matrix of order n.

Returns the n by n array with entries h[i,j] = 1 / (i + j + 1).

n : int
The size of the array to create.
h : (n, n) ndarray
The Hilbert matrix.

invhilbert : Compute the inverse of a Hilbert matrix.

New in version 0.10.0.

```>>> from scipy.linalg import hilbert
>>> hilbert(3)
array([[ 1.        ,  0.5       ,  0.33333333],
[ 0.5       ,  0.33333333,  0.25      ],
[ 0.33333333,  0.25      ,  0.2       ]])
```
`invhilbert`(n, exact=False)

Compute the inverse of the Hilbert matrix of order n.

The entries in the inverse of a Hilbert matrix are integers. When n is greater than 14, some entries in the inverse exceed the upper limit of 64 bit integers. The exact argument provides two options for dealing with these large integers.

n : int
The order of the Hilbert matrix.
exact : bool, optional
If False, the data type of the array that is returned is np.float64, and the array is an approximation of the inverse. If True, the array is the exact integer inverse array. To represent the exact inverse when n > 14, the returned array is an object array of long integers. For n <= 14, the exact inverse is returned as an array with data type np.int64.
invh : (n, n) ndarray
The data type of the array is np.float64 if exact is False. If exact is True, the data type is either np.int64 (for n <= 14) or object (for n > 14). In the latter case, the objects in the array will be long integers.

hilbert : Create a Hilbert matrix.

New in version 0.10.0.

```>>> from scipy.linalg import invhilbert
>>> invhilbert(4)
array([[   16.,  -120.,   240.,  -140.],
[ -120.,  1200., -2700.,  1680.],
[  240., -2700.,  6480., -4200.],
[ -140.,  1680., -4200.,  2800.]])
>>> invhilbert(4, exact=True)
array([[   16,  -120,   240,  -140],
[ -120,  1200, -2700,  1680],
[  240, -2700,  6480, -4200],
[ -140,  1680, -4200,  2800]], dtype=int64)
>>> invhilbert(16)[7,7]
4.2475099528537506e+19
>>> invhilbert(16, exact=True)[7,7]
42475099528537378560L
```
`pascal`(n, kind="symmetric", exact=True)

Returns the n x n Pascal matrix.

The Pascal matrix is a matrix containing the binomial coefficients as its elements.

n : int
The size of the matrix to create; that is, the result is an n x n matrix.
kind : str, optional
Must be one of ‘symmetric’, ‘lower’, or ‘upper’. Default is ‘symmetric’.
exact : bool, optional
If exact is True, the result is either an array of type numpy.uint64 (if n < 35) or an object array of Python long integers. If exact is False, the coefficients in the matrix are computed using scipy.special.comb with exact=False. The result will be a floating point array, and the values in the array will not be the exact coefficients, but this version is much faster than exact=True.
p : (n, n) ndarray
The Pascal matrix.

invpascal

New in version 0.11.0.

```>>> from scipy.linalg import pascal
>>> pascal(4)
array([[ 1,  1,  1,  1],
[ 1,  2,  3,  4],
[ 1,  3,  6, 10],
[ 1,  4, 10, 20]], dtype=uint64)
>>> pascal(4, kind='lower')
array([[1, 0, 0, 0],
[1, 1, 0, 0],
[1, 2, 1, 0],
[1, 3, 3, 1]], dtype=uint64)
>>> pascal(50)[-1, -1]
25477612258980856902730428600L
>>> from scipy.special import comb
>>> comb(98, 49, exact=True)
25477612258980856902730428600L
```
`invpascal`(n, kind="symmetric", exact=True)

Returns the inverse of the n x n Pascal matrix.

The Pascal matrix is a matrix containing the binomial coefficients as its elements.

n : int
The size of the matrix to create; that is, the result is an n x n matrix.
kind : str, optional
Must be one of ‘symmetric’, ‘lower’, or ‘upper’. Default is ‘symmetric’.
exact : bool, optional
If exact is True, the result is either an array of type numpy.int64 (if n <= 35) or an object array of Python integers. If exact is False, the coefficients in the matrix are computed using scipy.special.comb with exact=False. The result will be a floating point array, and for large n, the values in the array will not be the exact coefficients.
invp : (n, n) ndarray
The inverse of the Pascal matrix.

pascal

New in version 0.16.0.

 [1] “Pascal matrix”, http://en.wikipedia.org/wiki/Pascal_matrix
 [2] Cohen, A. M., “The inverse of a Pascal matrix”, Mathematical Gazette, 59(408), pp. 111-112, 1975.
```>>> from scipy.linalg import invpascal, pascal
>>> invp = invpascal(5)
>>> invp
array([[  5, -10,  10,  -5,   1],
[-10,  30, -35,  19,  -4],
[ 10, -35,  46, -27,   6],
[ -5,  19, -27,  17,  -4],
[  1,  -4,   6,  -4,   1]])
```
```>>> p = pascal(5)
>>> p.dot(invp)
array([[ 1.,  0.,  0.,  0.,  0.],
[ 0.,  1.,  0.,  0.,  0.],
[ 0.,  0.,  1.,  0.,  0.],
[ 0.,  0.,  0.,  1.,  0.],
[ 0.,  0.,  0.,  0.,  1.]])
```

An example of the use of kind and exact:

```>>> invpascal(5, kind='lower', exact=False)
array([[ 1., -0.,  0., -0.,  0.],
[-1.,  1., -0.,  0., -0.],
[ 1., -2.,  1., -0.,  0.],
[-1.,  3., -3.,  1., -0.],
[ 1., -4.,  6., -4.,  1.]])
```
`dft`(n, scale=None)

Discrete Fourier transform matrix.

Create the matrix that computes the discrete Fourier transform of a sequence [1]_. The n-th primitive root of unity used to generate the matrix is exp(-2*pi*i/n), where i = sqrt(-1).

n : int
Size the matrix to create.
scale : str, optional
Must be None, ‘sqrtn’, or ‘n’. If scale is ‘sqrtn’, the matrix is divided by sqrt(n). If scale is ‘n’, the matrix is divided by n. If scale is None (the default), the matrix is not normalized, and the return value is simply the Vandermonde matrix of the roots of unity.
m : (n, n) ndarray
The DFT matrix.

When scale is None, multiplying a vector by the matrix returned by dft is mathematically equivalent to (but much less efficient than) the calculation performed by scipy.fftpack.fft.

New in version 0.14.0.

 [1] “DFT matrix”, http://en.wikipedia.org/wiki/DFT_matrix
```>>> from scipy.linalg import dft
>>> np.set_printoptions(precision=5, suppress=True)
>>> x = np.array([1, 2, 3, 0, 3, 2, 1, 0])
>>> m = dft(8)
>>> m.dot(x)   # Compute the DFT of x
array([ 12.+0.j,  -2.-2.j,   0.-4.j,  -2.+2.j,   4.+0.j,  -2.-2.j,
-0.+4.j,  -2.+2.j])
```

Verify that `m.dot(x)` is the same as `fft(x)`.

```>>> from scipy.fftpack import fft
>>> fft(x)     # Same result as m.dot(x)
array([ 12.+0.j,  -2.-2.j,   0.-4.j,  -2.+2.j,   4.+0.j,  -2.-2.j,
0.+4.j,  -2.+2.j])
```