Random Complex-Valued Hermitian (Covariance) Matrix Generator

This repository provides a simple Python function make_cplx_spd_matrix that generates a random complex-valued, Hermitian, and positive semi-definite (PSD) covariance matrix. This function is useful in scenarios involving complex-valued random variables, such as in signal processing and wireless communications.

Overview

The function make_cplx_spd_matrix creates a random Hermitian matrix by following these steps:

1. Generates a random complex matrix $A$.
2. Computes the Singular Value Decomposition (SVD) of $A^H A$ (where $A^H$ is the conjugate transpose of $A$).
3. Constructs a new matrix $C$ using the SVD components and a random scaling matrix to ensure the resulting matrix is positive semi-definite.

The resulting matrix $C$ is Hermitian and PSD, suitable for use in simulations or applications requiring complex covariance matrices.

Requirements

• Python 3.x
• numpy (for numerical operations)
• scipy (for linear algebra operations)

You can install the required packages with pip:

pip install numpy scipy

Function Definition

# Function to generate a random complex PSD matrix
def make_cplx_spd_matrix(dim, rng=np.random.default_rng(1234567)):
    # Generate a random complex matrix A
    A = rng.random([dim, dim]) + 1j * rng.random([dim, dim])

    # Perform Singular Value Decomposition (SVD) on A.T.conj() * A
    U, _, Vt = scipy.linalg.svd(np.dot(A.T.conj(), A), check_finite=False)

    # Construct the final matrix X using SVD components and random scaling
    X = np.dot(np.dot(U, 1.0 + np.diag(rng.random(dim))), Vt)

    # Return the complex PSD matrix
    return X

Parameters:

• dim: The dimension of the square matrix (i.e., the number of rows and columns).
• rng: The random number generator. By default, it uses np.random.default_rng(1234567) for reproducibility.

Returns:

• A complex-valued Hermitian positive semi-definite matrix of shape (dim, dim).

Related Function

There is a related function in the scikit-learn library: sklearn.datasets.make_spd_matrix. However, this function only generates real-valued symmetric matrices, not complex-valued Hermitian ones.

License

This project is licensed under the MIT License - see the LICENSE file for details.