MATH 689-601 Random Matrix Theory, spring 2017
Course information
- Instructors:
- Dr Michael Anshelevich, Blocker 533d,
manshel AT math.tamu.edu, Office hours M 9:30-11; I will also be available TF 9:30-11, but may have other students.
- Dr Gregory Berkolaiko, Blocker 625c,
berko AT math.tamu.edu, Office hours T 2-3, F 11-12.
- Lectures: TR 12:45-2:00, BLOC 628
- Office hours: TBA.
- Course description (first day handout)
- The course does not have a textbook. We will most closely follow Todd Kemp's notes and Terry Tao's book (with an earlier version linked from his page). Additional resources include the book by Anderson, Guionnet, and Zeitouni (also on the web), and notes by Benedek Valko and Manjunath Krishnapur.
Course Notes
The notes will be updated cumulatively.
Matlab demonstrations
Homework Assignments
- For practice only.
- (**) Exercise 4.7 from the notes.
- (*) Exercise 4.8 from the notes.
- (**) Exercise 5.3 from the notes.
- (***) Exercise 5.9 from the notes.
- Due Mar 30, target submission 4*
- (*) Let \(H\) be an \(N\times N\) real symmetric matrix. Prove
that \(U = \exp(iH)\) is unitary symmetric.
- (***) Find a simple derivation of the Haar measure on the
compact group \(\mathbb{U}(2)\).
- (*) Let \(Z\) be a complex matrix of full rank and \(T:
Z\mapsto U\) the operator of the Gram-Schmidt
orthonormalization. Show that \(T(VZ) = VT(Z)\) for any unitary
\(V\).
- (** to ****) Let \(X_N\) be a random matrix with an odd
\(N\) (can be GOE(N) or CUE(N), for example). Let \(\lambda_0(X)\) be its
median eigenvalue (or eigenphase in the case of CUE(N), chosen
in the range \(-\pi \leq \lambda \leq \pi\)). Study numerically the
distribution of \(\lambda_0(X)\); compare it with the
distribution of the median of i.i.d. random variables with
appropriate distribution; compare the concentration in the two
models; compare it with the theoretical results we obtained;
make conjectures; find references.
- (**) Exercise 3.7 from the notes.
- (*) Exercise 3.13 from the notes. Don't worry about the full rigor.
- (**) Exercise 3.17 from the notes.
- Due Mar 2, target submission 4*
- (*) Consider the two-element group generated by an antiunitary
\(T\) such that \(T^2=1\). It has two linear representations,
multiplication by \(1\) and by \(-1\). One may think that it
also has two Wigner co-representations,
\[ \rho_1(T): z \mapsto \overline{z} \]
and
\[ \rho_2(T): z \mapsto -\overline{z}. \]
Prove that these corepresentations are actually equivalent!
(You will first need to figure out what is equivalence of
corepresentations).
- (*) Prove that all eigenvalues of a self-dual quaternion
matrix have even multiplicity (this is called "Kramers'
degeneracy").
- (**) In the space of all self-dual quaternion matrices, what is
the co-dimension of the set of matrices with an eigenvalue of
multiplicity four (i.e. when two Kramers' pairs coincide)? What
is therefore the expected power of level repulsion in GSE?
- (**) Wigner surmise for the eigenvalue spacing distribution for
GOE (\(\beta=1\)), GUE (\(\beta=2\)) and GSE (\(\beta=4\)) is
\[ p(s) = C_\beta s^\beta e^{-a_\beta s^2}, \] where
\(C_\beta\) and \(a_\beta\) are some constants. Derive those
constants in each case from the condition that \(p(s)\) is a
probability density function with mean 1.
- (*) Exercise 3.2 from the notes.
- (**) Exercise 3.5 from the notes.
- Due Feb 16, target submission 4*.
Most exercises this week come from the notes.
- (**) 2.4.
- (* to **) 2.10. (*) for the complex case, (**) for both the complex and quaternionic cases. Update: sketch of a solution added to the notes.
- (** to ****) 2.11. (**) for the \(N = K\) case, (***) for the full answer in the general \(K\) case in terms of combinatorial objects, (****) for the full answer (the latter one not recommended except for combinatorics students).
- (*) 2.14.
- (*) 2.28.
- (*) The complexification of quaternions is the algebra
\[ \{q_0 + q_1 j_1 + q_2 j_2 + q_3 j_3 : q_i \in \mathbb{C}\}\]
According to Frobenius's theorem, it is either isomorphic to \(\mathbb{R}, \mathbb{C}\), or \(\mathbb{H}\), or is not a division algebra. Which is it, and why?
- Due Feb 2, target submission 3*
- (*) Let \(X\) be a random matrix with independent entries;
assume each entry's distribution is symmetric. Show that
\[ \mathbb{E} \left[ \frac{1}{N} \mathrm{Tr} \left(X^{2k+1}\right)
\right] = 0 \]
- (*) Work out the value of the normalization constant \(Z\) in the
expression for the joint density of GOE
\[ p(X) =
\frac{1}{Z}\exp\left(-\frac{N}{4}\mathrm{Tr}(X^2)\right). \]
- (*) Derive the joint density for the GUE.
- (** to ***) Prove that \(d(O X O ^T) = dX\), where
\(dX = dx_{11} dx_{12} \ldots dx_{22} dx_{23} \ldots\) and
\(O\) is an orthogonal matrix. Do it for \(2\times2\) matrices
(**) and then work out the argument for \(N\times N\) matrices
(***). One possibility for the second part is to prove it for
Givens/Jacobi rotations and then use the fact that they
generate all orthogonal matrices. But there are certainly
better ways of proof.
- (**) Prove that
\[ \int x^{2k+1} d\sigma = 0, \qquad \int x^{2k} d\sigma = c_k,
\]
where \(\sigma\) is the semicircle density and \(c_k\) are the
Catalan numbers. Hint: integrate by parts to get some
recursive relation.
- (*) Plot numerically the distribution on the complex plane
of the eigenvalues of non-symmetric Gaussian matrices. All
entries are i.i.d. normal variables (either real or complex). Hint:
use Matlab command histogram2; if your matrices are
real, you will see the so-called "Saturn effect" which
diminishes with \(N\); there is no such effect for complex matrices.
This file was last modified on Wednesday, 10-Jan-2024 15:58:25 CST.