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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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