×

Spectral (isotropic) manifolds and their dimension. (English) Zbl 1337.53010

Summary: A set of \(n\times n\) symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in \(\mathbb{R}^n\) is called spectral or isotropic. In this paper, we establish that every locally symmetric \(C^k\) submanifold \(\mathcal{M}\) of \(\mathbb{R}^n\) gives rise to a \(C^k\) spectral manifold for \(k\in \{2, 3,\dots,\omega\}\). An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of \(\mathcal{M}\) is derived. This work builds upon the results of Sylvester and Šilhavý and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Software:

LMIRank
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. J. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51 (1984), 699-728. · Zbl 1077.74507 · doi:10.1215/S0012-7094-84-05134-2
[2] F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. · Zbl 0966.49001 · doi:10.1007/978-1-4612-1394-9
[3] J. Dadok, On the C∞Chevalley’s theorem, Adv. Math. 44 (1982), 121-131. · Zbl 0521.22009 · doi:10.1016/0001-8708(82)90002-0
[4] A. Daniilidis, D. Drusvyatskiy, and A. S. Lewis, Orthogonal invariance and identifiability, SIAM J. Matrix Anal. Appl. 35 (2014), 580-698. · Zbl 1339.15007 · doi:10.1137/130916710
[5] A. Daniilidis, A. S. Lewis, J. Malick, and H. Sendov, Prox-regularity of spectral functions and spectral sets, J. Convex Anal. 15 (2008), 547-560. · Zbl 1152.15013
[6] A. Daniilidis, A. S. Lewis, J. Malick, and H. Sendov, Locally symmetric submanifolds lift to spectral manifolds, arxiv:1212.3936[math.OC]. · Zbl 1152.15013
[7] A. Daniilidis, J. Malick, and H. Sendov, On the structure of locally symmetric manifolds, J. Convex Anal. 22 (2015), 399-426. · Zbl 1319.53005
[8] M. P. Do Carmo, Riemannian Geometry, Birkhäuser, Boston, Inc., Boston, MA, 1992. · doi:10.1007/978-1-4757-2201-7
[9] U. Helmke and J. B. Moore, Optimization and Dynamical Systems, second edition, Springer, New York, 1996. · Zbl 0984.49001
[10] J. B. Hiriart-Urruty and D. Ye, Sensitivity analysis of all eigenvalues of a symmetric matrix, Numer. Math. 70 (1992), 45-72. · Zbl 0816.15016 · doi:10.1007/s002110050109
[11] T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976. · doi:10.1007/978-3-642-66282-9
[12] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, John Wiley & Sons, New York-London, 1963. · Zbl 0119.37502
[13] A. S. Lewis, Derivatives of spectral functions, Math. Oper. Res. 21 (1996), 576-588. · Zbl 0860.49017 · doi:10.1287/moor.21.3.576
[14] A. S. Lewis, Nonsmooth analysis of eigenvalues, Math. Program. 84 (1999), 1-24. · Zbl 0969.49006
[15] A. S. Lewis and H. Sendov, Twice differentiable spectral functions, SIAM J. Matrix Anal. Appl. 23 (2001), 368-386. · Zbl 1053.15004 · doi:10.1137/S089547980036838X
[16] R. Orsi, U. Helmke, and J. B. Moore, A Newton-like method for solving rank constrained linear matrix inequalities, Automatica J. IFAC 42 (2006), 1875-1882. · Zbl 1222.90032 · doi:10.1016/j.automatica.2006.05.026
[17] R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans.Amer. Math. Soc. 348 (1996) 1805-1838. · Zbl 0861.49015 · doi:10.1090/S0002-9947-96-01544-9
[18] H. Sendov, The higher-order derivatives of spectral functions, Linear Algebra Appl. 424 (2007), 240-281. · Zbl 1126.49039 · doi:10.1016/j.laa.2006.12.013
[19] M. Silhavý, Differentiability properties of isotropic functions, Duke Math. J. 104 (2000), 367-373. · Zbl 1077.74507 · doi:10.1215/S0012-7094-00-10431-0
[20] J. Sylvester, On the differentiability of O(n) invariant functions of symmetric matrices, Duke Math. J. 52 (1985), 475-483. · Zbl 0652.26013 · doi:10.1215/S0012-7094-85-05223-8
[21] J. A. Tropp, I. S. Dhillon, R. W. Heath, and T. Strohmer, Designing structured tight frames via an alternating projection method, IEEE Trans. Inform. Theory 51 (2005), 188-209. · Zbl 1288.94021 · doi:10.1109/TIT.2004.839492
[22] N. K. Tsing, M. K. W. Fan, and E. I. Verriest, On analyticity of functions involving eigenvalues, Linear Algebra Appl. 207 (1994), 159-180. · Zbl 0805.15022 · doi:10.1016/0024-3795(94)90009-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.