Eigendistribution

Eigendistribution, also known as spectral distribution or eigenvalue distribution, refers to the distribution of eigenvalues of a matrix or an operator. It provides a statistical or qualitative understanding of the spectral properties, especially when dealing with large matrices or operators in various fields such as mathematics, physics, and engineering.

In linear algebra, the eigenvalues of a matrix are scalar values that indicate the factors by which eigenvectors are scaled during a linear transformation. The eigendistribution describes how these eigenvalues are spread across the complex plane or real line, offering insights into the matrix's behavior. For random matrices, the eigendistribution often converges to a specific probability distribution as the matrix size increases, such as the Wigner semicircle law for certain symmetric matrices or the Marchenko–Pastur distribution for sample covariance matrices.

In functional analysis, particularly in the spectral theory of operators, eigendistribution helps in understanding the spectral measure associated with self-adjoint or normal operators. This measure captures the distribution of spectral values and plays a crucial role in quantum mechanics, signal processing, and other applications.

Methods to analyze eigendistribution include empirical spectral measures, Wigner semicircle law, limiting spectral distributions, and spectral density functions. These tools allow researchers to study the asymptotic behavior of spectra and to derive properties relevant to stability, dynamics, or randomness.

Overall, eigendistribution serves as a fundamental concept in examining the spectral characteristics of matrices and operators, with broad applications across theoretical and applied sciences.