Hauptseminar Advanced Matrix Analysis (MA602022)
Wintersemester 2011/12
Prof. Dr. Michael M. Wolf, Dr. David Reeb
- Place and date: Thursdays 16:15-17:45, seminar room 03.10.011, Registration
- Contents: * Majorization * Variational principles * Matrix norms * Operator convex and operator monotone functions * Positive and completely positive maps * Non-negative and stochastic matrices * Matrix perturbation theory * Matrix calculus * Matrix groups and semigroups * Fast matrix multiplication * Matrix polynomials and polynomial identities * Eigenvalue inequalities and Horn's conjecture
- Literature: The first five topics are covered in Bhatia "Matrix Analysis" (Springer) and Bhatia "Positive Definite Matrices" (Princeton). For the remaining topics special literature will be provided.
- Prerequisites: Linear Algebra 2 (MA1102)
- Participation in each seminar session is compulsory (schedule below)
News
- Please be prepared for our joint discussion of Matrix Norms in the seminar session on November 24. For the specific points to be discussed, see the drop-down box in the schedule below.
- If you are not able to get a hold of Bhatia's books, you can come by my office (03.12.040, Dr. David Reeb) and get my copy for xeroxing.
- If you have questions during your preparation, contact Dr. David Reeb.
Format of the Presentations
- Duration: approx. 75 minutes, plus time for questions/discussion
- Medium: blackboard (possibly overhead slides or Powerpoint, but then need to go slowly)
- Language: English or German
- In the drop-down boxes in the schedule below it is incidated which points should be covered at least in each presentation. The presenter may want to add other points at his/her choosing for the presentation.
Schedule and Presentation Topics
- Chapter II in Bhatia "Matrix Analysis" (Springer)
- The presentation should cover at least the following: * definition "majorized"/"(weakly) sub-majorized" * definition "doubly stochastic matrix" * possibly Exercises II.1.5 and/or II.1.7 * Thm. II.1.10(i)(iv) * Exercise II.1.12 as an example for majorization (and for unitary-/ortho-stochasticity), possibly Exercise II.1.13 * Birkhoff's Theorem (Thm. II.2.3, Hall Theorem II.2.1) * Theorem II.3.1 * possibly Equation (II.23)
- As this topic is an important base for many of the following presentations, all participants are expected to read chapter IV in Bhatia's "Matrix Analysis" and should be prepared to contribute to a joint discussion of the main definitions, ideas and theorems during the seminar session.
- Chapter IV in Bhatia "Matrix Analysis" (Springer)
- Topics to be discussed include: * norms on C^n, including definition * connection to symmetric gauge functions * work out Example IV.1.4 * dual of a norm (Equation (IV.21)), show that this defines a norm, work out Exercise IV.1.13, Problem IV.5.4 * work out Exercise IV.1.15 explicitly for p=1 and for p=2 * definition "matrix norm", "sub-multiplicative norm", "unitarily invariant norm" * correspondence with symmetric gauge functions (Thm. IV.2.1, complete the proof) * operator norm, trace norm, Frobenius norm, Schatten p-norms, Ky Fan k-norms, induced norms (Problem IV.5.11) * (sketch proof that unitarily invariant norms are sub-multiplicative, pg. 94) * Hilbert-Schmidt inner product (Equation IV.35), dual matrix norm (pg. 96) * weakly unitarily invariant norms, numerical radius
- Chapter V in Bhatia "Matrix Analysis" (Springer)
- The presentation should cover at least the following: * Löwner's partial order * definition "operator monotonicity", "operator convexity" * examples in Section V.1, Proposition V.1.6, Theorem V.1.9 * Sketch proof idea and results from: Theorem V.2.3, Theorem V.2.5, Corollary V.2.6, Theorem V.2.9, Theorem V.2.10, Exercise V.2.11, Exercise V.2.13 * Löwner's Theorems (Corollary V.4.5, Theorem V.4.6; outline basic ideas)
- Bhatia "Positive Definite Matrices" (Princeton), up to page 71 (middle)
- The presentation should cover at least the following: * What is a positive (semi-)definite matrix? * What is a positive map? * What is a completely positive map? * One example of a positive map that is not completely positive. * The three basic representation theorems for completely positive maps: Kraus (Thm. 3.1.1), Stinespring Dilation (Thm. 3.1.2), Choi (Thm. 3.1.4, especially (i) and (iii)).
- [1] Horn&Johnson "Matrix Analysis" (1985), chapter 6. [2] Bhatia "Matrix Analysis", chapters VI-VIII. [3] Mackay, appendix C.3 (available online) ^{}
- The presentation should cover at least the following: * Gershgorin discs (6.1.1 in [1] + Corollaries) * condition number, perturbation theorems 6.3.1 or 6.3.2 in [1] * spectral perturbation of normal matrices, e.g. Hoffman/Wielandt-Theorem (6.3.5 in [1]; see also Thm VI.4.1 in [2]) or Weyl's Perturbation Theorem (Thm. VI.2.1 in [2]) * perturbation theory of non-degenerate eigenvalues and eigenvectors of non-normal matrices (6.3.10-6.3.12 in [1] with examples; see appendix C.3 in [3] for perturbation of eigenvectors)
- Other facets: * spectral perturbation of normal matrices (section VI.3 in [2]) * perturbation of eigenspaces of normal matrices (sections VII.3, VII.4 in [2]) * general spectral variation bounds (VIII.1, VIII.2, VI.1 in [2])
- [1] Boij&Laksov: Matrix Groups. [2] Prof. Wolf: Section 7.1.1 here. [3] Engel&Nagel: "A short course on operator semigroups" (available online) ^{}
- The presentation should cover at least the following: ***(1) Matrix groups ([1], specializing to groups over the real or complex numbers, i.e. K=R,C): * matrix groups over the complex/real numbers (section 1.1) * GL(n), SL(n), O(n), SO(n), Sp(n), U(n), SU(n) (section 1.1) * preservation of bilinear forms, classical groups (section 1.8) * exponential map and tangent spaces (esp. sections 2.2, second part of 2.3, 2.5, maybe 2.6, possibly 2.7) ***(2) One-parameter semigroups of matrices ([2]; see also [3], specializing to the finite-dimensional case): exposition in [2], some examples adapted from [3]
- [1] Chapter X.4 in Bhatia "Matrix Analysis" (Springer). [2] J. Dieudonne: "Foundations of Modern Analysis" (chapter VIII). [3] Ambrosetti&Prodi: "A Primer of Nonlinear Analysis" (chapters 1-2)
- Frechet derivative * Gauteaux derivative * rules of calculus * derivatives of some matrix functions * mean value theorem * higher derivatives * Taylor expansion * possibly some elements from chapter V.3 in [1]
- This seminar session has been cancelled.