# Differentiability of eigenvalue and eigenvector on the non-simple case

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.

• Symmetric matrices are diagonalizable, so how would it occur that A or B had a defective eigenvalue? – macbeth Oct 31 '16 at 23:42
• I replaced defective by non-simple – Shake Baby Oct 31 '16 at 23:49
• If you require that $\dim \ker J_h(x)$ is independent of $x$ then you get the differentiability you seek. – Liviu Nicolaescu Nov 1 '16 at 10:09
• Do you have a reference? – Shake Baby Nov 8 '16 at 18:08

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 $T(x)$.

• Thanks, but if I'm note mistaken, this holds only when $n=1$. – Shake Baby Oct 31 '16 at 23:46
• Ahh yes, you are right. Actually, the $n>1$ case seems to be more complicated - a remark in Kato's book a few page before the above mentioned theorem [on page 120, Remark 6.3], indicates that there is a difference between $n=1$ and $n=2$ already in the case of holomorphic matrix functions. Hope you will get the real answer soon! – Zoltan Zimboras Nov 1 '16 at 7:32

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.

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 $t$.

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.

• Thank you. This was very helpful. I'm reading about the local blow up, but the concept is very un-familiar to me. Can I recover the standard derivative of $A(x)$ by computing derivatives after the local blow up of the coordinates? – Shake Baby Nov 9 '16 at 3:10