SciPy - linalg.hilbert() Function

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The linalg.hilbert function in SciPy is used to create the Hilbert matrix of order n.

What is Hilbert Matrix?

Hilbert Matrix is a square matrix with entries being the unit fractions. These matrices are notoriously ill-conditioned, meaning that small changes in the input can lead to large changes in the solution when solving linear systems. This illustrates as an example of numerical instability in linear algebra computation.

The Hilbert matrix of a specified order n, where each element h[i,j] is calculated as −

Hij = 1⁄i + j + 1

Where, i and j are the row and column, respectively, starting from 1.

Syntax

Following is the syntax of the linalg.hilbert function in SciPy which is used to create Hilbert matrix −

scipy.linalg.hilbert(n)

Parameters

Following are the parameters of the scipy.linalg.hilbert() function −

• n − This parameter takes an integer data type as an input which represents the size of the array to be created.

Return Value

The scipy linalg.hilbert() function takes the above parameter and returns (n,n) ndarray that represents the Hilbert matrix .

Example 1

The following example uses scipy linalg.hilbert() function to create Hilbert matrix, where the parameter 'n' which represents the sixe of the array is passed with a positive integer.

from scipy.linalg import hilbert
hilbert(8)

Following is the output of the above program −

array([[1.        , 0.5       , 0.33333333, 0.25      , 0.2       ,
        0.16666667, 0.14285714, 0.125     ],
       [0.5       , 0.33333333, 0.25      , 0.2       , 0.16666667,
        0.14285714, 0.125     , 0.11111111],
       [0.33333333, 0.25      , 0.2       , 0.16666667, 0.14285714,
        0.125     , 0.11111111, 0.1       ],
       [0.25      , 0.2       , 0.16666667, 0.14285714, 0.125     ,
        0.11111111, 0.1       , 0.09090909],
       [0.2       , 0.16666667, 0.14285714, 0.125     , 0.11111111,
        0.1       , 0.09090909, 0.08333333],
       [0.16666667, 0.14285714, 0.125     , 0.11111111, 0.1       ,
        0.09090909, 0.08333333, 0.07692308],
       [0.14285714, 0.125     , 0.11111111, 0.1       , 0.09090909,
        0.08333333, 0.07692308, 0.07142857],
       [0.125     , 0.11111111, 0.1       , 0.09090909, 0.08333333,
        0.07692308, 0.07142857, 0.06666667]])

Example 2

Further, in this example the scipy.linalg.hilbert() function is used by passing the parameter 'n' with 0.

from scipy.linalg import hilbert
hilbert(0)

Output for the above code is as follows −

array([], shape=(0, 0), dtype=float64)

Example 3

In the following example code, we use the scipy.linalg.hilbert() function by passing the parameter 'n' with a negative integer.

from scipy.linalg import hilbert
hilbert(-100)

Following is the output of the −

array([], shape=(0, 0), dtype=float64)