Abstract
We prove that a positive matrix with all permutation products equal is diagonally equivalent to J, the all-1s matrix. Then we give a simple proof of the rank inequality for diagonally magic matrices (J. Inequal. Appl. 2015:318, 2015).
Highlights
We denote by Cn×n and Rn×n the sets of n × n complex matrices and n × n real matrices, respectively
A matrix A = ∈ Cn×n is called diagonally magic if n n ai,σ (i) = ai,π(i) for all σ, π ∈ Sn
Let C = ∈ Rn×n be a positive matrix with n n ci,γ (i) = cj,τ (j) for all γ, τ ∈ Sn
Summary
We denote by Cn×n and Rn×n the sets of n × n complex matrices and n × n real matrices, respectively. For a positive integer n, let Sn be the set of all n! Let A ∈ Cn×n, ≤ i ≤ i ≤ · · · ≤ ik ≤ n, and ≤ j ≤ j ≤ · · · ≤ js ≤ n. A matrix A = (ai,j) ∈ Cn×n is called diagonally magic if n n ai,σ (i) = ai,π(i) for all σ , π ∈ Sn. Obviously, the zero matrix n×n and J = [ ]n×n, the matrix of all ones, are diagonally magic matrices.
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