matrix_ops.vectors_to_gram_matrix

Calculates the Gram matrix from a list of vectors.

Functions

vectors_to_gram_matrix(vectors)

Construct the Gram matrix from a list of vectors [1].

Module Contents

matrix_ops.vectors_to_gram_matrix.vectors_to_gram_matrix(vectors)

Construct the Gram matrix from a list of vectors [1].

The Gram matrix is a matrix of inner products, where the entry G[i, j] is the inner product of vectors[i] and vectors[j]. This function computes the Gram matrix for a given list of vectors.

Examples

>>> # Example with real vectors:
>>> import numpy as np
>>> from toqito.matrix_ops import vectors_to_gram_matrix
>>> vectors = [np.array([1, 2]), np.array([3, 4])]
>>> gram_matrix = vectors_to_gram_matrix(vectors)
>>> gram_matrix
array([[ 5, 11],
       [11, 25]])

>>> # Example with complex vectors:
>>> import numpy as np
>>> from toqito.matrix_ops import vectors_to_gram_matrix
>>> vectors = [np.array([1+1j, 2+2j]), np.array([3+3j, 4+4j])]
>>> gram_matrix = vectors_to_gram_matrix(vectors)
>>> gram_matrix
array([[10.+0.j, 22.+0.j],
       [22.+0.j, 50.+0.j]])

References

[1] (1,2)

Wikipedia. Gram matrix. URL: https://en.wikipedia.org/wiki/Gram_matrix.

Raises:

ValueError – If the vectors are not all of the same length.

Parameters:

vectors (list[numpy.ndarray]) – A list of vectors (1D numpy arrays). All vectors must be of the same length.

Returns:

A list of vectors corresponding to the ensemble of states.

Return type:

numpy.ndarray