Trace of Multi-variable Matrix Functions and its Application to Functions of Graph Spectrum
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arXiv
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| Format: | Preprint |
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2025
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| author | Bhattacharya, Subhrajit |
| author_facet | Bhattacharya, Subhrajit |
| contents | 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 $g(M) = \text{Tr}(f(M)) = \sum_{j=1}^n f(μ_j)$ (where $\{μ_j\}_{j=1,2,\cdots,n}$ 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 $g(M) = \sum_{j_1=1}^n \sum_{j_2=1}^n \cdots \sum_{j_m=1}^n f(μ_{j_1}, μ_{j_2},\cdots, μ_{j_m})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14515 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trace of Multi-variable Matrix Functions and its Application to Functions of Graph Spectrum Bhattacharya, Subhrajit Functional Analysis Combinatorics Operator Algebras 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 $g(M) = \text{Tr}(f(M)) = \sum_{j=1}^n f(μ_j)$ (where $\{μ_j\}_{j=1,2,\cdots,n}$ 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 $g(M) = \sum_{j_1=1}^n \sum_{j_2=1}^n \cdots \sum_{j_m=1}^n f(μ_{j_1}, μ_{j_2},\cdots, μ_{j_m})$ 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. |
| title | Trace of Multi-variable Matrix Functions and its Application to Functions of Graph Spectrum |
| topic | Functional Analysis Combinatorics Operator Algebras |
| url | https://arxiv.org/abs/2501.14515 |