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Lower bounds for the smallest singular value. (English) Zbl 0986.15016
The author presents a lower bound formula for the smallest singular value in the case of an $$n\times n$$ complex matrix $$A=(a_{ij})$$ satisfying the condition: for any $$i\in \{ 1,2,\dots,n\}$$ there exists $$j\not=i$$ such that at least one of $$a_{ij}\not=0$$ and $$a_{ji}\not=0$$ holds. The lower bound is expressed in terms of the numbers $$r_k=\sum_{j\not=k}|a_{kj}|$$ and $$c_k=\sum_{j\not=k}|a_{jk}|$$, and the presented result simplifies and improves that of C. R. Johnson and T. Szulc [Linear Algebra Appl. 272, 169-179 (1998; Zbl 0891.15013)].

##### MSC:
 15A42 Inequalities involving eigenvalues and eigenvectors 15A18 Eigenvalues, singular values, and eigenvectors
##### Keywords:
smallest singular value; undirected graph; lower bound
Full Text:
##### References:
 [1] Johnson, C.R; Szulc, T, Further lower bounds for the smallest singular value, Linear algebra and its applications, 272, 169-179, (1998) · Zbl 0891.15013 [2] Horn, R.A; Johnson, C.R, Topics in matrix analysis, (1991), Cambridge University Press New York · Zbl 0729.15001 [3] Rojo, O; Soto, R; Rojo, H, Bounds for the spectral radius and the largest singular value, Computers math. applic., 36, 1, 41-50, (1998) · Zbl 0942.15013 [4] Horn, R.A; Johnson, C.R, Matrix analysis, (1985), Cambridge University Press New York · Zbl 0576.15001
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