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Problem

Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$

Find the eigenvalue distribution using whatever you can.

Background

In my field, I have a Bayesian inference framework that will obtain the $X$ distribution, but what we really need is the eigenvalue distribution of the matrix $X$.

Question

  1. Is this problem well defined?
  2. Is the only way to infer it through sampling in such high dimensional space and then for each realization of the matrix, do the eigenvalue decomposition?
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  • $\begingroup$ The problem is certainly well defined as the vector of eigenvalues is a function of the matrix. I lack the knowledge on the topic, but I think circular law might be interesting to you (I encourage you to check out random matrices). $\endgroup$ – Jakub Bartczuk Jun 26 '18 at 12:18
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I am not aware of any useful distributional result here, and my suspicion is that the distribution of the eigenvalues will be so complicated that you will have to simulate it through sampling. However, it is worth noting that eigenvalues are roots of the characteristic polynomial $p_\boldsymbol{X}(t) = \det (t \boldsymbol{I} - \boldsymbol{X})$, so what you are looking for is the distribution of the roots of a polynomial, with coefficients that are themselves complicated functions of underlying normal random variables.

There is a bit of mathematical and statistical literature on the distribution of roots of polynomials, and related root-based problems (see e.g., Erdos and Turan 1950, James 1960, James 1964, Bogomolny et al 1992, Kostlan 1993). I am nowhere near an expert in this area, but it seems to me that the general finding in this literature is that the distribution of polynomial roots is complicated even in cases with simple distributions for the coefficients. So in the case where you're looking for the roots of the characteristic polynomial of a random matrix with arbitrary normal elements, this is probably going to be crazy-difficult to deal with analytically.

The simplest case ($2 \times 2$ matrix): One useful thing to consider is how hard this problem is in the simplest case where you have a random matrix:

$$\boldsymbol{X} = \begin{bmatrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{bmatrix}.$$

In this case the eigenvalues have a well-known form. Letting $T = X_{11}+X_{22}$ be the trace of the random matrix, and $D = X_{11}X_{22}-X_{12}X_{21}$ the determinant, the eigenvalues are:

$$\lambda_1 = \tfrac{1}{2} \Big( T + \sqrt{T^2 - 4 D} \Big),$$ $$\lambda_2 = \tfrac{1}{2} \Big( T - \sqrt{T^2 - 4 D} \Big).$$

Since $D = \lambda_1 (T - \lambda_1) = \lambda_2 (T - \lambda_2)$, it is possible to derive the joint density of the eigenvalues using the marginal density of $T$ (which is normal) and the conditional density of $D|T$ (which is complicated). The latter distribution is complicated, so even in the simplest case of a $2 \times 2$ matrix, you have a joint density for the eigenvalues that will be an integral over a complicated conditional distribution. I can only imagine how complicated the result would be for larger matrices!

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