2023/09/26 by Einollahzadeh, Mostafa
#Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2309.14958
We obtain tight lower bounds for the trace norm \Vert ⋅ \Vert1 of some matrices with diagonal zero, in terms of the entry-wise L1-norm (denoted by \Vert ⋅ \Vert(1)). It is shown that on the space of nonzero real symmetric matrices A of order n with diagonal zero, the minimum value of the quantity \frac\Vert A\Vert1\Vert A\Vert(1) is equal to (2)/(n). The answer of the similar problem in the space of Hermitian matrices, is also obtained to be equal to tan(\fracπ2n). The equivalent "dual" form of these results, give some upper bounds for the distance to the nearest diagonal matrix for a given symmetric or Hermitian matrix, when the distance is computed in the spectral norm.