A note on the variation of permanents
\textit{R. Bhatia} [ibid. 62, 269-276 (1984; Zbl 0552.15014)] showed that for any two \(n\times n\)-matrices A, B the inequality \((*)\quad | per(A)-per(B)| \leq n\| A-B\|_ 2\max (\| A\|_ 2,\| B\|_ 2)^{n-1}\) holds. Using quite elementary methods, the author of the paper under review proves the analogous result for \(\|.\|_ 2\) being replaced by the row-sum and the column-sum norms \(\| A\|_{\infty}=\max (| a_{i1}|,...,| a_{in}|),\) respectively \(\| A\|_ 1=\max_{k}(| a_{1k}|,...,| a_{nk}|).\) He conjectures that (*) is still true if \(\|.\|_ 2\) is replaced by any operator norm for \(n\times n\)-matrices. In this case, a determinantal analogue has been settled by \textit{S. Friedland} [Linear Multilinear Algebra 12, 81-98 (1982; Zbl 0491.15002)].
scientific article; zbMATH DE number 4065173