Johnson's determinantal identity for contiguous minors of Toeplitz matrices, with an accretive extension

Abstract

Let $A$ be an $n\times n$ real Toeplitz matrix satisfying $A+A^{\top }=2{\mathbb{J}}_n$, where ${\mathbb{J}}_n$ is the all-ones matrix.If $A_r(i,j)$ denotes the $r\times r$ contiguous submatrix of $A$ consisting of rows $i,i+1,\dots ,i+r-1$ and columns $j,j+1,\dots ,j+r-1$, then for every $n\ge 2$ one has $$\det A_{n-1}(1,2)+\det A_{n-1}(2,1)=2\det A_{n-1}(1,1).$$ This confirms a conjecture of Charles R.~Johnson (2003). The proof combines a rank-one determinant expansion with Dodgson's condensation formula, and then invokes a polynomial-identity argument in the Toeplitz parameters: after obtaining an equality of squares in the integral domain $\mathbb{Z}[b_1,\dots ,b_{n-1}]$, we factor it to deduce an identity up to sign and determine the sign by a suitable specialization.We also give an extension of the Bayat--Teimoori arithmetic--geometric mean identity: for every real accretive matrix $A$, one has the sharp inequality $$\sqrt{\det A_{n-1}(1,1)\det A_{n-1}(2,2)}\ge \left|\frac{\det A_{n-1}(1,2)+\det A_{n-1}(2,1)}{2}\right|,$$ with equality whenever the symmetric part has rank one, i.e.\ $A+A^{\top }=\alpha w{w}^{\top }$ for some $\alpha \in \mathbb{R}$ and $w\in {\mathbb{R}}^n\setminus \{0\}$,recovering the Bayat--Teimoori equality as a special case.

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