On the Geometry of the Set of Symmetric Matrices with Repeated Eigenvalues

Arnold Mathematical Journal - Tập 4 Số 3-4 - Trang 423-443 - 2018
Paul Breiding1, Khazhgali Kozhasov2, Antonio Lerario3
1Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany
2Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
3SISSA, Trieste, Italy

Tóm tắt

Abstract We investigate some geometric properties of the real algebraic variety $$\Delta $$ Δ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart–Young–Mirsky-type theorem for the distance function from a generic matrix to points in $$\Delta $$ Δ . We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of $$\Delta $$ Δ ) and random matrix theory.

Từ khóa


Tài liệu tham khảo

Agrachev, A.A.: Spaces of symmetric operators with multiple ground states. Funktsional. Anal. i Prilozhen. 45(4), 1–15 (2011)

Agrachev, A.A., Lerario, A.: Systems of quadratic inequalities. Proc. Lond. Math. Soc. (3) 105(3), 622–660 (2012)

Arnold, V.I.: Topological properties of eigen oscillations in mathematical physics. Tr. Mat. Inst. Steklova 273(Sovremennye Problemy Matematiki), 30–40 (2011)

Arnold, V.I.: Modes and quasimodes. Funkcional. Anal. i Priložen. 6(2), 12–20 (1972)

Arnold, V.I.: Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect. Sel. Math. (N.S.) 1(1), 1–19 (1995)

Arnold, V.I.: Frequent representations. Mosc. Math. J. 3(4), 1209–1221 (2003)

Beltrán, C., Kozhasov, Kh.: The real polynomial eigenvalue problem is well conditioned on the average (2018). arXiv:1802.07493 [math.NA]

Beltrán, C.: Estimates on the condition number of random rank-deficient matrices. IMA J. Numer. Anal. 31(1), 25–39 (2011)

Ben Arous, G., Bourgade, P.: Extreme gaps between eigenvalues of random matrices. Ann. Probab. 41(4), 2648–2681 (2013)

Bik, A., Draisma, J.: A note on ED degrees of group-stable subvarieties in polar representations (2017). arXiv:1708.07696 [math.AG]

Breiding, P., Kozhasov, Kh., Lerario, A.: Random spectrahedra (2017). arXiv:1711.08253 [math.AG]

Bürgisser, P., Cucker, F.: Condition: The Geometry of Numerical Algorithms, Grundlehren der Mathematischen Wissenschaften, vol. 349. Springer, Heidelberg (2013)

Demmel, J.W.: The probability that a numerical analysis problem is difficult. Math. Comput. 50(182), 449–480 (1988)

Draisma, J., Horobet, E., Ottaviani, G., Sturmfels, B., Thomas, R.R.: The Euclidean distance degree of an algebraic variety. Found. Comput. Math. 16(1), 99–149 (2016)

Drusvyatskiy, D., Lee, H.-L., Ottaviani, G., Thomas, R.R.: The Euclidean distance degree of orthogonally invariant matrix varieties. Isr. J. Math. 221(1), 291–316 (2017)

Edelman, A., Kostlan, E.: How many zeros of a random polynomial are real? Bull. Am. Math. Soc. (N.S.) 32(1), 1–37 (1995)

Edelman, A., Kostlan, E., Shub, M.: How many eigenvalues of a random matrix are real? J. Am. Math. Soc. 7(1), 247–267 (1994)

Gradshteyn, I., Ryzhik, I.: Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam (2015)

Helmke, U., Shayman, M.A.: Critical points of matrix least squares distance functions. Linear Algebra Appl. 215, 1–19 (1995)

Howard, R.: The kinematic formula in Riemannian homogeneous spaces. Mem. Am. Math. Soc 106(509), vi+69 (1993)

Ilyushechkin, N.V.: Some identities for elements of a symmetric matrix. J. Math. Sci. 129(4), 3994–4008 (2005)

Kostlan, E.: On the expected number of real roots of a system of random polynomial equations. In: Foundations of computational mathematics (Hong Kong, 2000), pp 149–188. World Sci. Publ., River Edge (2002)

Lerario, A., Lundberg, E.: Gap probabilities and Betti numbers of a random intersection of quadrics. Discrete Comput. Geom. 55(2), 462–496 (2016)

Mehta, M .L.: Random matrices, 3rd edn. Elsevier, Amsterdam (2004)

Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1982)

Nguyen, H., Tao, T., Vu, V.: Random matrices: tail bounds for gaps between eigenvalues. Probab. Theory Relat. Fields 167(3–4), 777–816 (2017)

Parlett, B.N.: The (matrix) discriminant as a determinant. Linear Algebra Appl. 355(1), 85–101 (2002)

Sanyal, R., Sturmfels, B., Vinzant, C.: The entropic discriminant. Adv. Math. 244, 678–707 (2013)

Shapiro, M., Vainshtein, A.: Stratification of Hermitian matrices and the Alexander mapping. C. R. Acad. Sci. 321(12), 1599–1604 (1995)

Shub, M., Smale, S.: Complexity of Bezout’s theorem. III. Condition number and packing. J. Complex. 9(1), 4–14 (1993) (Festschrift for Joseph F. Traub, Part I. (1993))

Shub, M., Smale, S.: Complexity of Bezout’s theorem. II. Volumes and probabilities. In: Computational algebraic geometry (Nice, 1992), Progr. Math., vol. 109, pp. 267–285. Birkhäuser Boston, Boston (1993)

Shub, M., Smale, S.: Complexity of Bézout’s theorem. I. Geometric aspects. J. Am. Math. Soc 6(2), 459–501 (1993)

Spanier, J., Oldham, K.B., Myland, J.: An atlas of functions. Springer, Berlin (2000)

Tao, T.: Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)

Teytel, M.: How rare are multiple eigenvalues? Commun. Pure Appl. Math. 52(8), 917–934 (1999)

Vassiliev, V.A.: Spaces of Hermitian operators with simple spectra and their finite-order cohomology. Mosc. Math. J. 3(3), 1145–1165, 1202 (2003). (Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday)