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A note for bound of Euclidean norm of group inverse. (English) Zbl 1181.15028
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 \(A^\#\):
\[ \| A^\#\|_2\geq \sqrt{{n\over {2n\choose n}- 1}}. \]
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
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