Differentiability of eigenvalue and eigenvector on the non-simple case

Question

Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=J_h^\mathtt{T}(x)J_h(x)\in\mathbb{R}^{n\times n}$ and $B(x):=J_h(x)J_h(x)^\mathtt{T}\in\mathbb{R}^{m\times m}$.

I'm interested in sufficient conditions ensuring differentiability of the functions $U(x)$, $\Sigma(x)$ and $V(x)$ in a singular value decomposition of $J_h(x)=U(x)\Sigma(x)V(x)^\mathtt{T}$ when there is at least one repeating zero singular value (rank deficient case).

The question can be equivalently stated in terms of eigenvalues/eigenvectors of the symmetric matrices $A$ and $B$. Are there sufficient conditions to ensure differentiability of an eigenpair with a non-simple eigenvalue?

Appreciate any help.

Answer 1

I think Theorem 6.8 on page 122 in Kato: Perturbation Theory for Linear Operators may help (at least for the question concerning the eigenvalues of the symmetric $A$ and $B$ matrices).

Theorem: Assume that $T(x)$ is a symmetric and continuously differentiable ($N \times N$ matrix) function in an interval $I$ of $x$. Then there exist $N$ continuously differentiable functions $\mu_n(x)$ on $I$ that represent the repeated eigenvalues of $T(x)$.

Answer 2

Let me point out a more specific result for hyperbolic polynomials, known as Bronshtein's theorem (see e.g. the preprint https://arxiv.org/abs/1309.2150 by A. Parusinski & A. Rainer). Let $p(X,y)$ be a polynomial with degree $m$ in the $X$ variable depending smoothly on $y\in \mathbb R^n$ and assume that the roots $\{\lambda_j(y)\}_{1\le j\le m}$ are real-valued (this is the hyperbolicity condition). Then a Lipschitz-continuous choice of the $\lambda_j$ is possible.

The improvement due to hyperbolicity is striking since without that assumption, Hölderian regularity with index $1/m$ is the best we can hope. The preprint quoted above is nicely written and is shedding new light on a classical result whose original proof was not so easily available.

Answer 3

For symmetric matrices, you are good (in fact even in infinite dimension). Let me quote the MathReview of the following reference (itself quoting the article):

Kriegl, Andreas(A-WIEN); Michor, Peter W.(A-ERS) Differentiable perturbation of unbounded operators. (English summary) Math. Ann. 327 (2003), no. 1, 191–201.

Theorem. Let $t\mapsto A(t)$ for $t\in\Bbb R$ be a curve of unbounded self-adjoint operators in a Hilbert space with common domain of definition and with compact resolvent. If $A(t)$ is real analytic in $t\in\Bbb R$, then the eigenvalues and the eigenvectors of $A(t)$ may be parameterized real analytically in $t$.

I guess several parameters does not hurt, and that it should be simpler for matrices. I also guess that Peter Michor will be able to say more, he is quite active on MO.

Let me mention that of course, when there is multiplicity and without self-adjointness one is in bad shape: you could find matrices with characteristic polynomials having factors like $x^2-t$ whose eigenvalue would not depend smoothly on $t$.

Answer 4

Suppose $h$ is real analytic. Then $A(x)$ and $B(x)$ are real analytic in $x\in \mathbb R^n$. Part L of the main theorem of

• Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011), 407-416. [(pdf)][1]

shows that the eigenvalues of the symmetric matrix valued functions $A$ and $B$ can be chosen real analytic in $x$ after a local blow up of the coordinates $x$.

If $h$ is smooth you need further assumptions. They are spelled out in this paper.