A note for bound of Euclidean norm of group inverse. (English) Zbl 1181.15028

Yalçiner, Aynur; Taşkara, Necati

Int. J. Contemp. Math. Sci. 4, No. 9-12, 519-524 (2009).

For a finite square complex matrix \(A= [a_{ij}]^n_{i,j=1}\), its Euclidean norm is \[ \| A\|_2= \Biggl(\sum^n_{i,j= 1}|a_{ij}|^2\Biggr)^{1/2} \] and its group inverse is the unique matrix \(A^\#\) satisfying \(AA^\# A= A\), \(A^\# AA^\#= A^\#\) and \(AA^\#= A^\# A\). In this note, the authors obtain, for the circulant matrix \(A= \left[{n\choose\text{mod}(j-i,n)}\right]^n_{i,j=1}\), a lower bound for the Euclidean norm of its group inverse \(A^\#\):
\[ \| A^\#\|_2\geq \sqrt{{n\over {2n\choose n}- 1}}. \]

Reviewer: Pei Yuan Wu (Hsinchu)

MSC:

15A45	Miscellaneous inequalities involving matrices
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A09	Theory of matrix inversion and generalized inverses

Keywords:

circulant matrix; Euclidean norm; lower bound; group inverse