Trace of Multi-variable Matrix Functions and its Application to Functions of Graph Spectrum

Matrix extension of a scalar function of a single variable is well-studied in literature. Of particular interest is the trace of such functions. It is known that for diagonalizable matrices, $M$, the function $g(M)=\text{Tr}(f(M))={\sum }_{j=1}^{n}f({\mu }_j)$ (where $\{{\mu }_j{\}}_{j=1,2,\cdots ,n}$ are the eigenvalues of $M$) inherits the monotonocity and convexity properties of $f$ (i.e., for $g$ to be convex, $f$ need not be operator convex – convexity is sufficient). In this paper we formalize the idea of matrix extension of a function of multiple variables, study the monotonicity and convexity properties of the trace, and thus show that a function of form $g(M)={\sum }_{j_1=1}^n{\sum }_{j_2=1}^n\cdots {\sum }_{j_m=1}^{n}f({\mu }_{j_1},{\mu }_{j_2},\cdots ,{\mu }_{j_m})$ also inherits the monotonocity and convexity properties of the multi-variable function, $f$. We apply these results to functions of the spectrum of the weighted Laplacian matrix of undirected, simple graphs.