MATH 689-601 Random Matrix Theory, spring 2017

Course information

Course Notes

The notes will be updated cumulatively.

Matlab demonstrations

Homework Assignments

  1. For practice only.
    1. (**) Exercise 4.7 from the notes.
    2. (*) Exercise 4.8 from the notes.
    3. (**) Exercise 5.3 from the notes.
    4. (***) Exercise 5.9 from the notes.
  2. Due Mar 30, target submission 4*
    1. (*) Let \(H\) be an \(N\times N\) real symmetric matrix. Prove that \(U = \exp(iH)\) is unitary symmetric.
    2. (***) Find a simple derivation of the Haar measure on the compact group \(\mathbb{U}(2)\).
    3. (*) 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 \(V\).
    4. (** 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.
    5. (**) Exercise 3.7 from the notes.
    6. (*) Exercise 3.13 from the notes. Don't worry about the full rigor.
    7. (**) Exercise 3.17 from the notes.
  3. Due Mar 2, target submission 4*
    1. (*) 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).
    2. (*) Prove that all eigenvalues of a self-dual quaternion matrix have even multiplicity (this is called "Kramers' degeneracy").
    3. (**) 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?
    4. (**) 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.
    5. (*) Exercise 3.2 from the notes.
    6. (**) Exercise 3.5 from the notes.
  4. Due Feb 16, target submission 4*. Most exercises this week come from the notes.
    1. (**) 2.4.
    2. (* to **) 2.10. (*) for the complex case, (**) for both the complex and quaternionic cases. Update: sketch of a solution added to the notes.
    3. (** 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).
    4. (*) 2.14.
    5. (*) 2.28.
    6. (*) 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?
  5. Due Feb 2, target submission 3*
    1. (*) Let \(X\) be a random matrix with independent entries; assume each entry's distribution is symmetric. Show that \[ \mathbb{E} \left[ \frac{1}{N} \mathrm{Tr} \left(X^{2k+1}\right) \right] = 0 \]
    2. (*) Work out the value of the normalization constant \(Z\) in the expression for the joint density of GOE \[ p(X) = \frac{1}{Z}\exp\left(-\frac{N}{4}\mathrm{Tr}(X^2)\right). \]
    3. (*) Derive the joint density for the GUE.
    4. (** 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.
    5. (**) 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.
    6. (*) 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.

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