Matrix Chernoff bound

For certain applications in linear algebra, it is useful to know properties of the probability distribution of the largest eigenvalue of a finite sum of random matrices. Suppose ${\displaystyle \{\mathbf {X} _{k}\}}$ is a finite sequence of random matrices. Analogous to the well-known Chernoff bound for sums of scalars, a bound on the following is sought for a given parameter t:

${\displaystyle \Pr \left\{\lambda _{\max }\left(\sum _{k}\mathbf {X} _{k}\right)\geq t\right\}}$

The following theorems answer this general question under various assumptions; these assumptions are named below by analogy to their classical, scalar counterparts. All of these theorems can be found in (Tropp 2010), as the specific application of a general result which is derived below. A summary of related works is given.