toqito.rand.random_circulant_gram_matrix¶

Generates a random circulant Gram matrix.

Module Contents¶

toqito.rand.random_circulant_gram_matrix.random_circulant_gram_matrix(dim, seed=None)[source]¶

Generate a random circulant Gram matrix of specified dimension.

A circulant matrix is a square matrix where the elements of each row are identical to the elements of the previous row such that the elements in one row are relocated by 1 position (in a cyclic manner) compared to the previous row. The eigenvalues and eigenvectors of this matrix are derived from the Discrete Fourier Transform (DFT).

For more information on circulant matrices, see [@WikiCirculantMat]. This function utilizes the normalized DFT, a variation of DFT with normalized basis vectors.

For additional information, see [@DSPNormDFT].

The function creates a circulant matrix from a random diagonal matrix and the normalized DFT matrix. First, it generates a diagonal matrix with random non-negative entries. Next, it constructs the normalized DFT matrix. Finally, it computes the circulant matrix, which is real due to its origin from the DFT of a real diagonal matrix.

Examples

Generate a random circulant Gram matrix of dimension 4.

```python exec=”1” source=”above” session=”random_circulant_gram_matrix_example” import numpy as np from toqito.rand import random_circulant_gram_matrix

circulant_matrix = random_circulant_gram_matrix(4)

print(f”Shape of circulant matrix is {circulant_matrix.shape}”) ```

`python exec="1" source="above" session="random_circulant_gram_matrix_example" print(np.allclose(circulant_matrix, circulant_matrix.T)) `

`python exec="1" source="above" session="random_circulant_gram_matrix_example" print(circulant_matrix) `

It is also possible to pass a seed to this function for reproducibility.

```python exec=”1” source=”above” from toqito.rand import random_circulant_gram_matrix

circulant_matrix = random_circulant_gram_matrix(4, seed=42)

print(circulant_matrix) ```

Parameters:

dim (int) – int The dimension of the circulant matrix to generate.
seed (int | None) – int | None A seed used to instantiate numpy’s random number generator.

Returns:

numpy.ndarray A dim x dim real, symmetric, circulant matrix.

Return type:

numpy.ndarray