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#### Keywords

Equiradial class; Spectral norm; Singular values; Gersgorin set.

#### Abstract

In estimating the largest singular value in the class of matrices equiradial with a given $n$-by-$n$ complex matrix $A$, it was proved that it is attained at one of $n(n-1)$ sparse nonnegative matrices (see C.R.~Johnson, J.M.~Pe{\~n}a and T.~Szulc, Optimal Gersgorin-style estimation of the largest singular value; {\em Electronic Journal of Linear Algebra Algebra Appl.}, 25:48--59, 2011). Next, some circumstances were identified under which the set of possible optimizers of the largest singular value can be further narrowed (see C.R.~Johnson, T.~Szulc and D.~Wojtera-Tyrakowska, Optimal Gersgorin-style estimation of the largest singular value, {\it Electronic Journal of Linear Algebra Algebra Appl.}, 25:48--59, 2011). Here the cardinality of the mentioned set for $n$-by-$n$ matrices is further reduced. It is shown that the largest singular value, in the class of matrices equiradial with a given $n$-by-$n$ complex matrix, is attained at one of $n(n-1)/2$ sparse nonnegative matrices. Finally, an inequality between the spectral radius of a $3$-by-$3$ nonnegative matrix $X$ and the spectral radius of a modification of $X$ is also proposed.

#### Recommended Citation

Johnson, Charles R.; Pena, J. M.; and Szulc, Tomasz.
(2016),
"Optimal Gersgorin-style estimation of the largest singular value. II",
*Electronic Journal of Linear Algebra*,
Volume 31, pp. 679-685.

DOI: https://doi.org/10.13001/1081-3810.3033

*Abstract*