DifferentiableEigen.jl

The current implementation of LinearAlgebra.eigen does not support sensitivities. This package adds a new function eigen, that wraps the original function, but returns an array of reals instead of complex numbers (this is necessary, because some AD-frameworks do not support complex numbers). This eigen function is differentiable by every AD-framework with support for ChainRulesCore.jl and ForwardDiff.jl.

1. Open a Julia-REPL, switch to package mode using ], activate your preferred environment.

2. Install DifferentiableEigen.jl:

(@v1.6) pkg> add DifferentiableEigen

3. If you want to check that everything works correctly, you can run the tests bundled with DifferentiableEigen.jl:

(@v1.6) pkg> test DifferentiableEigen

import DifferentiableEigen
import LinearAlgebra
import ForwardDiff

A = rand(3,3)   # Random matrix 3x3 

eigvals, eigvecs = LinearAlgebra.eigen(A)   # This is the default eigen-function in Julia. Note, that eigenvalues and -vectors are complex numbers.
jac = ForwardDiff.jacobian((A) -> LinearAlgebra.eigen(A)[1], A)   # That doesn't work!

eigvals, eigvecs = DifferentiableEigen.eigen(A)   # This is the differentiable eigen-function. Note, that eigenvalues and -vectors are not complex numbers, but real arrays!  
jac = ForwardDiff.jacobian((A) -> DifferentiableEigen.eigen(A)[1], A)   # That does work! eigenvalue- and eigenvector-sensitvities

This package was motivated by this discourse thread. For now, there is no other (known) ready to use solution for differentiable eigenvalues and -vectors. If this changes, please feel free to open a PR or discussion.

The sensitivity formulas are picked from:

Michael B. Giles. 2008. An extended collection of matrix derivative results for forward and reverse mode algorithmic differentiation. PDF

Tobias Thummerer and Lars Mikelsons. 2023. Eigen-informed NeuralODEs: Dealing with stability and convergence issues of NeuralODEs. ArXiv. PDF