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From Eigenvalues to Singular Values: A Review ()

The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.

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*Advances in Pure Mathematics*, Vol. 3 No. 9B, 2013, pp. 8-24. doi: 10.4236/apm.2013.39A2002.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
R. Bhatia, “Matrix Analysis,” Springer, New York, 1997. http://dx.doi.org/10.1007/978-1-4612-0653-8 |

[2] | X. Chen and W. Li, “On the Rayleigh Quotient for Singular Values,” Journal of Computational Mathematic, Vol. 25, 2007, pp. 512-521. |

[3] |
A. Dax, “On Extremum Properties of Orthogonal Quotient Matrices,” Linear Algebra and its Applications, Vol. 432, No. 5, 2010, pp. 1234-1257. http://dx.doi.org/10.1016/j.laa.2009.10.034 |

[4] |
J. W. Demmel, “Applied Numerical Linear Algebra,” SIAM, Philadelphia, 1997. http://dx.doi.org/10.1137/1.9781611971446 |

[5] |
G. Eckart and G. Young, “The Approximation of One Matrix by Another of Lower Rank,” Psychometrika, Vol. 1, No. 3, 1936, pp. 211-218. http://dx.doi.org/10.1007/BF02288367 |

[6] |
A. Edelman, T. Arias and S. Smith, “The Geometry of Algorithms with Orthogonality Constraints,” The SIAM Journal on Matrix Analysis and Applications, Vol. 20, No. 2, 1998, pp. 303-353. http://dx.doi.org/10.1137/S0895479895290954 |

[7] |
A. Edelman and S. Smith, “On Conjugate Gradient-Like Methods for Eigen-Like Problems,” BIT Numerical Mathematics, Vol. 36, No. 3, 1996, pp. 494-508. http://dx.doi.org/10.1007/BF01731929 |

[8] |
L. Elden, “Matrix Methods in Data Mining and Pattern Recognition,” SIAM, Philadelphia, 2007. http://dx.doi.org/10.1137/1.9780898718867 |

[9] | L. Elden and H. Park, “A Procrust S Problem on the Stiefel Manifold,” Technical Report, Department of Mathematics, Linkoping University, Linkoping, 1997. |

[10] | K. Fan, “On a theorem of Weyl Concerning Eigenvalues of Linear Transformations I,” Proceedings of the National Academy of Sciences, USA, Vol. 35, No. 11, 1949, pp. 652-655. http://dx.doi.org/10.1073/pnas.35.11.652 |

[11] |
K. Fan, “Maximum Properties and Inequalities for the Eigenvalues of Completely Continuous Operators,” Proceedings of the National Academy of Sciences, USA, Vol. 37, No. 11, 1951, pp. 760-766. http://dx.doi.org/10.1073/pnas.37.11.760 |

[12] |
K. Fan, “A Minimum Property of the Eigenvalues of a Hermitian Transformation,” The American Mathematical Monthly, Vol. 60, No. 1, 1953, pp. 48-50. http://dx.doi.org/10.2307/2306486 |

[13] | E. Fischer, “Concerning Quadratic Forms with Real Coefficients,” Monatshefte für Mathematik und Physik, Vol. 16, 1906, pp. 234-249. |

[14] |
G. H. Golub and C. Van Loan, “An Analysis of the Total Least Squares Problem,” SIAM Journal on Numerical Analysis, Vol. 17, No. 6, 1980, pp. 883-893. http://dx.doi.org/10.1137/0717073 |

[15] | G. H. Golub and C. F. Van Loan, “Matrix Computations,” Johns Hopkins University Press, Baltimore, 1983. |

[16] |
R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cambridge University Press, 1985. http://dx.doi.org/10.1017/CBO9780511810817 |

[17] |
R. A. Horn and C. R. Johnson, “Topics in Matrix Analysis,” Cambridge University Press, Cambridge, 1991. http://dx.doi.org/10.1017/CBO9780511840371 |

[18] | R. M. Johnson, “On a Theorem Stated by Eckart and Young,” Psychometrica, Vol. 28, No. 3, 1963, pp. 259-263. http://dx.doi.org/10.1007/BF02289573 |

[19] | W. Kahan, B. N. Parlett and E. Jiang, “Residual Bounds on Approximate Eigensystems of Nonnormal Matrices,” SIAM Journal on Numerical Analysis, Vol. 19, No. 3, 1982, pp. 470-484. |

[20] |
R.-C. Li, “On Eigenvalues of a Rayleigh Quotient Matrix,” Linear Algebra and Its Applications, Vol. 169, 1992, pp. 249-255. http://dx.doi.org/10.1016/0024-3795(92)90181-9 |

[21] | X. G. Liu, “On Rayleigh Quotient Theory for the Eigenproblem and the Singular Value Problem,” Journal of Computational Mathematics, Supplementary Issue, 1992, pp. 216-224. |

[22] | A. W. Marshall and I. Olkin, “Inequalities: Theory of Majorization and Its Applications,” Academic Press, New York, 1979. |

[23] |
L. Mirsky, “Symmetric Gauge Functions and Unitarily Invariant Norms,” Quarterly Journal of Mathematics, Vol. 11, No. 1, 1960, pp. 50-59. http://dx.doi.org/10.1093/qmath/11.1.50 |

[24] | J. von Neumann, “Some Matrix-Inequalities and Metrization of Matrix-Space,” Tomsk. Univ. Rev., Vol. 1, 1937, pp. 286-300. (In: A. H. Taub, Ed., John von Neumann Collected Works, Vol. IV, Pergamon, Oxford, 1962, pp. 205-218.) |

[25] | D. P. O’Leary and G. W. Stewart, “On the Convergence of a new Rayleigh Quotient Method with Applications to Large Eigenproblems,” TR-97-74, Institute for Advanced Computer Studies, University of Maryland, College Park, 1997. |

[26] |
A. M. Ostrowski, “On the Convergence of the Rayleigh Quotient Iteration for the Computation of the Characteristic Roots and Vectors. III (Generalized Rayleigh Quotient and Characteristic Roots with Linear Elementary Divisors,” Archive for Rational Mechanics and Analysis, Vol. 3, No. 1, 1959, pp. 325-340. http://dx.doi.org/10.1007/BF00284184 |

[27] |
A. M. Ostrowski, “On the Convergence of the Rayleigh Quotient Iteration for the Computation of the Characteristic Roots and Vectors. IV (generalized Rayleigh Quotient for Nonlinear Elementary Divisors,” Archive for Rational Mechanics and Analysis, Vol. 3, No. 1, 1959, pp. 341-347. http://dx.doi.org/10.1007/BF00284185 |

[28] | M. L. Overton and R. S. Womersley, “On the Sum of the Largest Eigenvalues of a Symmetric Matrix,” SIAM Journal on Numerical Analysis, Vol. 13, No. 1, 1992, pp. 41-45. http://dx.doi.org/10.1137/0613006 |

[29] |
B. N. Parlett, “The Rayleigh Quotient Iteration and Some Generalizations for Nonnormal Matrices,” Mathematics of Computation, Vol. 28, No. 127, 1974, pp. 679-693. http://dx.doi.org/10.1090/S0025-5718-1974-0405823-3 |

[30] | B. N. Parlett, “The Symmetric Eigenvalue Problem,” Prentice-Hall, Englewood Cliffs, 1980. |

[31] |
E. Schmidt, “Zur Theorie der Linearen und Nichtlinearen Integralgleichungen. I Teil. Entwicklung Willkürlichen Funktionen nach System Vorgeschriebener,” Mathematische Annalen, Vol. 63, No. 4, 1907, pp. 433-476. http://dx.doi.org/10.1007/BF01449770 |

[32] | G. W. Stewart, “Two Simple Residual Bounds for the Eigenvalues of Hermitian Matrices,” SIAM Journal on Matrix Analysis and Applications, Vol. 12, No. 2, 1991, pp. 205-208. http://dx.doi.org/10.1137/0612016 |

[33] | G. W. Stewart, “Matrix Algorithms,” Vol. I: Basic Decompositions, SIAM, Philadelphia, 1998. |

[34] | G. W. Stewart, “Matrix Algorithms,” Vol. II: Eigensystems, SIAM, Philadelphia, 2001. |

[35] | E. Stiefel, “Richtungsfelder und Fernparallelismus in n-Dimensionalen Mannigfaltigkeiten,” Commentarii Mathematici Helvetici, Vol. 8, No. 1, 1935-1936, pp. 305-353. |

[36] | J. G. Sun, “Eigenvalues of Rayleigh Quotient Matrices,” Numerische Mathematik, Vol. 59, No. 1, 1991, pp. 603-614. http://dx.doi.org/10.1007/BF01385798 |

[37] | R. C. Thompson, “Principal Submatrices IX: Interlacing Inequalities for Singular Values of Submatrices,” Linear Algebra and its Applications, Vol. 5, 1972, pp. 1-12. |

[38] |
R. C. Thompson, “The Behavior of Eigenvalues and Singular Values under Perturbations of Restricted Rank,” Linear Algebra and Its Applications, Vol. 13, No. 1-2, 1976, pp. 69-78. http://dx.doi.org/10.1016/0024-3795(76)90044-6 |

[39] | H. Weyl, “Das Asymptotische Verteilungsgesetz der Eigenwerte Linearer Partieller Differentialgleichungen (Mit Einer Anwendung auf die Theorie der Hohlraumstrahlung,” Mathematische Annalen, Vol. 71, No. 4, 1912, pp. 441-479. http://dx.doi.org/10.1007/BF01456804 |

[40] | J. H. Wilkinson, “The Algebraic Eigenvalue Problem,” Clarendon Press, Oxford, 1965. |

[41] |
F. Zhang, “Matrix Theory: Basic Results and Techniques,” Springer-Verlag, New York, 1999. http://dx.doi.org/10.1007/978-1-4757-5797-2 |

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