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Bounds for norms of the matrix inverse and the smallest singular value

✍ Scribed by Nenad Morača

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
172 KB
Volume
429
Category
Article
ISSN
0024-3795

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✦ Synopsis

In the first part, we obtain two easily calculable lower bounds for A -1 , where • is an arbitrary matrix norm, in the case when A is an M-matrix, using first row sums and then column sums. Using those results, we obtain the characterization of M-matrices whose inverses are stochastic matrices. With different approach, we give another easily calculable lower bounds for A -1 ∞ and A -1 1 in the case when A is an M-matrix. In the second part, using the results from the first part, we obtain our main result, an easily calculable upper bound for A -1 1 in the case when A is an SDD matrix, thus improving the known bound. All mentioned norm bounds can be used for bounding the smallest singular value of a matrix.

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Further lower bounds for the smallest si
Further lower bounds for the smallest singular value
✍ Charles R. Johnson; Tomasz Szulc 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 464 KB

In an earlier paper of the first author, Gersgorin's theorem was used in a novel way to give a simple lower bound for the smallest singular value of a general complex matrix. That lower bound was stronger than previous published bounds. Here, we use three variants of Gersgorin's theorem in a similar