Understanding Hermitian Matrices and the Numpy.linalg.eigvalsh() Function

Do you work in the fields of quantum mechanics, electrical engineering, or signal processing? If so, you may come across Hermitian matrices in your work.

Hermitian matrices have many useful properties that make them a useful tool in a variety of applications. In this article, we will explore the definition, applications, and implementation of Hermitian matrices, as well as how to calculate their eigenvalues using the numpy.linalg.eigvalsh() function.

## Definition of Hermitian Matrices

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, if A is a Hermitian matrix, then A = A, where A is the conjugate transpose of A.

Conjugate transpose means taking the transpose of the matrix and then taking the complex conjugate of each element. Hermitian matrices have some unique properties.

Firstly, all of their eigenvalues are real numbers. Secondly, their eigenvectors are not only orthogonal but also form a complete basis for the space.

Finally, the magnitude of each eigenvalue corresponds to the degree to which the linear transformation associated with the matrix stretches or compresses the eigenvectors.

## Applications of Hermitian Matrices

As we’ve briefly mentioned, Hermitian matrices are used frequently in quantum mechanics, electrical engineering, and signal processing. For example, in quantum mechanics, Hermitian matrices are used to represent observables.

In electrical engineering, Hermitian matrices can represent resistance matrices. In signal processing, they are often used to represent correlation matrices.

## Implementing and Manipulating Hermitian Matrices

Numerical methods such as eigenvalue decomposition and singular value decomposition are commonly used to manipulate Hermitian matrices. For example, eigenvalue decomposition is used to find the eigenvectors and corresponding eigenvalues of a matrix.

Singular value decomposition is used to find the singular values, left-singular vectors, and right-singular vectors of a matrix. Using Numpy.linalg.eigvalsh() to Calculate Eigenvalues

Numpy is a popular Python library for scientific computing.

The numpy.linalg.eigvalsh() function can be used to calculate the eigenvalues of a Hermitian matrix. This function is specifically designed for Hermitian matrices, so you don’t have to worry about providing the function with a non-Hermitian matrix.

## Calculating Eigenvalues of Hermitian Matrices

Now that we have a basic understanding of Hermitian matrices and their function in various applications, let’s take a closer look at how to calculate their eigenvalues using Numpy’s eigvalsh() function.

## Properties of Hermitian Matrix Eigenvalues

We know that all of a Hermitian matrix’s eigenvalues are real numbers, but what else should we know about them? All Hermitian matrix eigenvalues have unique values, so there are no repeated eigenvalues.

Also, the determinant of a Hermitian matrix is equal to the product of all of its eigenvalues.

## Calculating Eigenvalues of Hermitian Matrices

To calculate the eigenvalues of a Hermitian matrix using Numpy’s eigvalsh() function, you simply need to provide the function with the matrix as input. The function will then return the eigenvalues sorted in ascending order.

The numpy.linalg.eigvalsh() function uses a variety of numerical algorithms to calculate the eigenvalues of a matrix. However, it’s worth mentioning that sometimes the algorithm may fail to converge.

In these cases, the function raises a LinAlgError.

Example 1: Calculating Eigenvalues of a Predefined Complex Hermitian Matrix

Let’s walk through an example of how to use numpy.linalg.eigvalsh() to calculate the eigenvalues of a predefined complex Hermitian matrix:

A = np.array([[1+0j, 5+15j, -4+3j], [5-15j, 61+0j, -37+2j], [-4-3j, -37-2j, 14+0j]])

w, v = np.linalg.eigvalsh(A)

## print(w)

In this example, we define a 3×3 Hermitian matrix A by providing a nested list of complex numbers. We then call numpy.linalg.eigvalsh() with A as input.

The result is a NumPy array of the eigenvalues sorted in ascending order. The output for this example is: [-35.24602736, -2.74637838, 81.99240574]

Example 2: Calculating Eigenvalues of a 3×3 Hermitian Matrix

Let’s look at a second example of how to calculate the eigenvalues of a 3×3 Hermitian matrix:

A = np.array([[2, -1, 1], [-1, 2, -1], [1, -1, 2]])

w, v = np.linalg.eigvalsh(A)

## print(w)

In this example, we define a 3×3 Hermitian matrix A by providing a nested list of real numbers. We then call numpy.linalg.eigvalsh() with A as input.

The result is an array of the eigenvalues. The output for this example is: [1.38196601, 2., 2.61803399]

## Conclusion

In this article, we learned about the properties of Hermitian matrices, their applications in various fields, and how to calculate their eigenvalues using Numpy’s eigvalsh() function. We walked through two examples, one involving a complex Hermitian matrix and one involving a real Hermitian matrix, to illustrate how to use the eigvalsh() function.

As you continue to use Hermitian matrices in your work, remember that their unique properties can help you solve a variety of problems. In this article, we explored the definition, applications, and implementation of Hermitian matrices, and how to calculate their eigenvalues using Numpy’s eigvalsh() function.

We discussed the unique properties of Hermitian matrices, their applications in various fields, and how to manipulate them using numerical methods such as eigenvalue decomposition and singular value decomposition. We also provided two examples to illustrate how to use the eigvalsh() function to calculate eigenvalues of Hermitian matrices.

Understanding Hermitian matrices and their properties is important for many fields, and using Numpy’s eigvalsh() function can simplify the process of calculating their eigenvalues.