The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex $n \times n$ Gaussian monic polynomials degree $k$. obtain exact formulae almost sure limit ESD in two distinct scenarios: (1) \rightarrow \infty$ $k$ constant (2) $k $n$ constant. The main tool our approach is replacement principle by Tao, Vu Krishnapur. Along way, also develop some auxiliary results potential independent interest: slightly extend result B\"{u}rgisser Cucker on tail bound norm pseudoinverse non-zero mean matrix, several estimates singular values certain structured matrices.