Column ranks and their preservers over nonnegative real matrices

Beasley, LeRoy B.; Song, Seok-Zun; Kang, Kyung‐Tae; Sarma, B. K.

doi:10.1016/j.laa.2004.04.022

Cited by 7 publications

(7 citation statements)

References 9 publications

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“…, G nn are one-by-one with non-overlapping row (column) indices. Since r(G jj ) = 1, we see that G jj = [1] for all j = 1, . .…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…the skew-symmetric matrix f (B)−f (B) tr can have nonzero entries only in (1, 2), (1, 3), (2, 1), (3,1) positions, and the absolute value of these entries cannot exceed µ. On the other hand, W (f (B)) = W (B), which is known to be the circular disk centered at zero with radius µ/ √ 2 (see [16] or [11,Theorem 4.1], for example), and therefore ±iµ √ 2 are eigenvalues of the matrix f (B) − f (B)…”

Section: Assertion 42mentioning

confidence: 99%

“…It follows that the (1, 2), (2, 1), (1,3), (3,1) entries of f (B) − f (B) tr have absolute values equal to µ, and f (B) must be one of the following four matrices:…”

Section: Assertion 42mentioning

confidence: 99%

“…The purpose of this paper is to characterize preservers of the spectral radius, numerical radius, or spectral norm of the sum of nonnegative matrices. There are not many works in the literature on preservers in the context of real entrywise nonnegative matrices: we mention [18], where spectrum preservers are described, [1,10,24,25] that deal with column rank preservers; [2] is concerned with primitivity preservers, and in [4] preserver problems that have to do with irreducibility are considered. In all these works, the linearity of the map f is assumed.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Maps preserving spectral radius, numerical radius, spectral norm

Li¹,

Poon²,

Rastogi³

2007

ELA

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Let M + n be the set of entrywise nonnegative n × n matrices. Denote by r(A) the spectral radius (Perron root) of A ∈ M + n . Characterization is obtained for maps f : M + n → M + n such that r(f (A) + f (B)) = r(A + B) for all A, B ∈ M + n . In particular, it is shown that such a map has the formfor some S ∈ M + n with exactly one positive entry in each row and each column. Moreover, the same conclusion holds if the spectral radius is replaced by the spectrum or the peripheral spectrum. Similar results are obtained for maps on the set of n × n nonnegative symmetric matrices. Furthermore, the proofs are extended to obtain analogous results when spectral radius is replaced by the numerical range, or the spectral norm. In the case of the numerical radius, a full description of preservers of the sum is also obtained, but in this case it turns out that the standard forms do not describe all such preservers.

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“…, G nn are one-by-one with non-overlapping row (column) indices. Since r(G jj ) = 1, we see that G jj = [1] for all j = 1, . .…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

Section: Assertion 42mentioning

confidence: 99%

“…It follows that the (1, 2), (2, 1), (1,3), (3,1) entries of f (B) − f (B) tr have absolute values equal to µ, and f (B) must be one of the following four matrices:…”

Section: Assertion 42mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Maps preserving spectral radius, numerical radius, spectral norm

Li¹,

Poon²,

Rastogi³

2007

ELA

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show abstract

“…It is necessary to note that there are several rank functions over a semiring that are analogues of the classical function of the matrix rank over a field. Detailed research and self-contained information about rank functions over semirings can be found in [1] and [5].…”

Section: Introductionmentioning

confidence: 99%