A combinatorial proof of the trace Cayley-Hamilton theorem
Sudip Bera
TL;DR
The paper develops a graph-theoretic framework to reinterpret the characteristic polynomial and trace invariants of a matrix $A$ via a weighted digraph $\mathcal{D}(A)$. It shows that coefficients $d_i$ of $p_A(\lambda)$ relate to sign-weighted sums over linear subdigraphs and that $\text{Tr}(A^k)$ equals the total weight of closed walks of length $k$, establishing a direct combinatorial bridge between principal minors, LSDs, and traces. The main contribution is a complete combinatorial proof of the Trace Cayley–Hamilton identities, achieved through a sign-reversing involution that cancels contributions in two regimes: $r>n$ and $1\le r\le n$. This graphical interpretation provides explicit trace relations and deepens the connection between algebraic invariants and graph-theoretic structures.
Abstract
The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed graph D(A), where the algebraic behavior of A is reflected in the combinatorial properties of D(A). In particular, the determinant and characteristic polynomial of A admit elegant formulations in terms of sign-weighted sums over linear subdigraphs of D(A), thereby providing a graphical interpretation of fundamental algebraic quantities. Building upon this correspondence, we establish a combinatorial proof of the trace Cayley-Hamilton theorem. This theorem furnishes explicit trace identities linking the coefficients of the characteristic polynomial of A with the traces of its successive powers.