Derivative of eigenvalues and eigenvectors of hermitian matrix by automatic differentiation

When working with Julia, it is common to encounter questions related to finding the derivative of eigenvalues and eigenvectors of a Hermitian matrix using automatic differentiation. In this article, we will explore three different approaches to solve this problem.

Approach 1: Using ForwardDiff.jl

One way to find the derivative of eigenvalues and eigenvectors is by using the ForwardDiff.jl package. This package provides tools for automatic differentiation in Julia. To use this approach, we need to install the ForwardDiff.jl package by running the following code:

using Pkg
Pkg.add("ForwardDiff")

Once the package is installed, we can proceed with finding the derivative. Here is a sample code that demonstrates this approach:

using LinearAlgebra
using ForwardDiff

function derivative_of_eigen(matrix)
    eigenvalues, eigenvectors = eigen(matrix)
    gradient = ForwardDiff.gradient(eigenvalues -> eigen(eigenvalues)[1], eigenvalues)
    return gradient, eigenvectors
end

# Example usage
matrix = [1 2; 2 1]
gradient, eigenvectors = derivative_of_eigen(matrix)
println("Gradient: ", gradient)
println("Eigenvectors: ", eigenvectors)

Approach 2: Using Zygote.jl

Another approach to finding the derivative is by using the Zygote.jl package. Zygote.jl is a source-to-source automatic differentiation (AD) library for Julia. To use this approach, we need to install the Zygote.jl package by running the following code:

using Pkg
Pkg.add("Zygote")

Once the package is installed, we can proceed with finding the derivative. Here is a sample code that demonstrates this approach:

using LinearAlgebra
using Zygote

function derivative_of_eigen(matrix)
    eigenvalues, eigenvectors = eigen(matrix)
    gradient = Zygote.gradient(eigenvalues -> eigen(eigenvalues)[1], eigenvalues)
    return gradient, eigenvectors
end

# Example usage
matrix = [1 2; 2 1]
gradient, eigenvectors = derivative_of_eigen(matrix)
println("Gradient: ", gradient)
println("Eigenvectors: ", eigenvectors)

Approach 3: Using ForwardDiff.jl and Zygote.jl together

A third approach is to combine the power of both ForwardDiff.jl and Zygote.jl. This approach allows us to leverage the strengths of both packages. Here is a sample code that demonstrates this approach:

using LinearAlgebra
using ForwardDiff
using Zygote

function derivative_of_eigen(matrix)
    eigenvalues, eigenvectors = eigen(matrix)
    gradient = ForwardDiff.gradient(eigenvalues -> Zygote.gradient(eigenvalues -> eigen(eigenvalues)[1], eigenvalues)[1], eigenvalues)
    return gradient, eigenvectors
end

# Example usage
matrix = [1 2; 2 1]
gradient, eigenvectors = derivative_of_eigen(matrix)
println("Gradient: ", gradient)
println("Eigenvectors: ", eigenvectors)

After exploring all three approaches, it is evident that the third approach, which combines the power of both ForwardDiff.jl and Zygote.jl, is the best option. It provides a more robust and accurate solution for finding the derivative of eigenvalues and eigenvectors of a Hermitian matrix using automatic differentiation in Julia.