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Linear spaces of Toeplitz and nilpotent matrices

1993
*
Journal of combinatorial theory. Series A
*

We also give elementary proofs

doi:10.1016/0097-3165(93)90025-4
fatcat:c442rshjbrbgbdifzqatryol6y
*of*theorems*of*Gerstenhaber concerning the dimension*of**linear**spaces**of**nilpotent**matrices*. ... Our proofs exhibit*and*exploit a combinatorial structure*of*these*linear**spaces*. ... The classes we consider are*spaces**of**Toeplitz**matrices*, circulant*matrices*,*and**nilpotent**matrices*. ...##
###
Report on the Dublin matrix theory conference, March 1984

1985
*
Linear Algebra and its Applications
*

Undoubtedly, the most extensively studied infinite

doi:10.1016/0024-3795(85)90221-6
fatcat:jqdxntdfzndkboxoyggwexq764
*Toeplitz*(*and*Hankel)*matrices*are those that give rise to operators on*spaces**of*squaresummable sequences;*and**of*these, far*and*away the most interesting ... ny nonzero entries*and*subspaces*of*the*space**of*all complex sequences. In some instances, they determine continuous*linear*transformations-called operators-between certain normed sequence*spaces*. ...##
###
Isometries of the Toeplitz Matrix Algebra
[article]

2015
*
arXiv
*
pre-print

We study the structure

arXiv:1502.01573v1
fatcat:66xti5ng7ja47cajgfhg7gp3k4
*of*isometries defined on the algebra A*of*upper-triangular*Toeplitz**matrices*. ... In our second result we use a range*of*ideas in operator theory*and**linear*algebra to show that every*linear*isometry A→ M_n(C) is*of*the form A UAV where U*and*V are two unitary*matrices*. ... Theorem 6.2 above is predicted by a theorem*of*Blecher*and*Labuschagne [6, Corollary 2.5 (3) ]. ...##
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Products of commuting nilpotent operators

2007
*
The Electronic Journal of Linear Algebra
*

*Matrices*that are products

*of*two (or more) commuting square-zero

*matrices*

*and*

*matrices*that are products

*of*two commuting

*nilpotent*

*matrices*are characterized. ... vector

*space*that are products

*of*two commuting

*nilpotent*operators. http://math.technion.ac.il/iic/ela types (i)

*and*(iii) if s is odd. ... Let V be an infinite-dimensional vector

*space*

*and*A : V → V a

*nilpotent*operator with index

*of*

*nilpotency*n. ...

##
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Page 2830 of Mathematical Reviews Vol. , Issue 2004d
[page]

2004
*
Mathematical Reviews
*

These n x n

*Toeplitz**matrices*stabilize the canonical principal*nilpotent*matrix. ... The marquee result*of*this paper is a parametrization*of*the*space**of*totally positive lower triangular n x n*Toeplitz**matrices*by their lower left corner minors. ...##
###
On some properties of Toeplitz matrices

2016
*
Cogent Mathematics
*

We also give some results regarding circulant

doi:10.1080/23311835.2016.1154705
fatcat:4fm4d3vlr5hyri7kre3sxuhkbe
*matrices*, skew-circulant*matrices**and*approximation by*Toeplitz**matrices*over the field*of*complex numbers. ... In this paper, we investigate some properties*of**Toeplitz**matrices*with respect to different matrix products. ... Involutory*and**nilpotent**Toeplitz**matrices*over the finite field ℤ p Let (n, p)*and*(n, p), respectively, denote the number*of*involutory*Toeplitz**matrices**and**nilpotent**Toeplitz**matrices**of*degree two ...##
###
Patterned linear systems: Rings, chains, and trees

2010
*
49th IEEE Conference on Decision and Control (CDC)
*

In this second paper we study canonical patterns: rings, chains,

doi:10.1109/cdc.2010.5717515
dblp:conf/cdc/HamiltonB10
fatcat:xsequz77krd7hmsls5xd7isopy
*and*trees,*and*we give examples drawn from multiagent systems, cellular chemistry,*and*control*of*diffusion processes. ... In a first paper we studied system theoretic properties*of*patterned systems*and*solved classical control synthesis problems with the added requirement to preserve the system pattern. ... It is easily shown that N n = 0, so N is*nilpotent**and*we call N the fundamental*nilpotent*matrix. It is easily shown that every lower triangular*Toeplitz*matrix is a function*of*N. ...##
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Efficient matrix preconditioners for black box linear algebra

2002
*
Linear Algebra and its Applications
*

We present new conditioners, including conditioners to preserve low displacement rank for

doi:10.1016/s0024-3795(01)00472-4
fatcat:4uc75rjqerborfo2bito2yn7jy
*Toeplitz*-like*matrices*. ... We also provide new analyses*of*preconditioner performance*and*results on the relations among preconditioning problems*and*with*linear*algebra problems. ... Introduction In the black box approach [15] one takes an external view*of*a matrix: it is a*linear*operator on a vector*space*. ...##
###
Page 5980 of Mathematical Reviews Vol. , Issue 92k
[page]

1992
*
Mathematical Reviews
*

Let {E};, E\2,---, E,,} be the standard basis

*of*the*linear**space**of*k xk real*matrices*. ... Our results are reasonably complete in the triangular case,*and*preliminary in the cases*of*nontriangu- lar*Toeplitz**matrices*, block*Toeplitz**matrices*,*and**Toeplitz*-like*matrices*with smoothly varying ...##
###
Is every matrix similar to a Toeplitz matrix?

1999
*
Linear Algebra and its Applications
*

The proof is constructive,

doi:10.1016/s0024-3795(99)00131-7
fatcat:uffivojszzewfcm6x6cj2y5vla
*and*can be adapted to nonderogatory*matrices*with entries in any field*of*characteristic zero or characteristic greater than n. ... We also prove that every n × n complex matrix with n ≤ 4 is similar to a*Toeplitz*matrix. ... But the set*of*compact operators is a two-sided ideal in the algebra*of*bounded*linear*operators on a Hilbert*space*[13, p. 85] , so every operator similar to a compact operator must itself be compact ...##
###
Representation spaces of the Jordan plane
[article]

2012
*
arXiv
*
pre-print

We investigate relations between the properties

arXiv:1209.0746v1
fatcat:l6fo3q47zvam5kpvhn2gjbdanu
*of*an algebra*and*its varieties*of*finite-dimensional module structures, on the example*of*the Jordan plane R=k/ (xy-yx-y^2). ... The influence*of*the property*of*the non-commutative Koszul (Golod-Shafarevich) complex to be a DG-algebra resolution*of*an algebra (NCCI), on the structure*of*representation*spaces*is studied. ... Let us present the*linear**space*U T n*of*upper triangular n × n*matrices*as the direct sum*of*two subspaces U T n = L 1 ⊕ L 2 , where L 1 consists*of**matrices*with zeros on upper diagonals with numbers ...##
###
Irreducible components of the Jordan varieties
[article]

2012
*
arXiv
*
pre-print

Along this line we establish an analogue

arXiv:0903.3820v3
fatcat:egsmt544wre53aixmskmob2zli
*of*the Gerstenhaber--Taussky--Motzkin theorem on the dimension*of*algebras generated by two commuting*matrices*. ... We show that all image algebras*of*n-dimensional representations are tame for n ≤ 4*and*wild for n ≥ 5. ... Let us present the*linear**space*U T n*of*upper triangular n × n*matrices*as the direct sum*of*two subspaces U T n = L 1 ⊕ L 2 , where L 1 consists*of**matrices*with zeros on upper diagonals with numbers ...##
###
An operator not satisfying Lomonosov's hypothesis

1980
*
Journal of Functional Analysis
*

An example is presented

doi:10.1016/0022-1236(80)90073-7
fatcat:bm4nszvbe5ew7fi4oaszmzmrdm
*of*a Hilbert*space*operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. ... The invariant subspace theorem*of*Lomonosov [6-S] includes the following assertion: if C is an operator such that CB = BC for an operator B that is not a multiple*of*the identity*and*that commutes with ... (M:) to AA, '*and*hence, by Lemma 3, is*nilpotent**of*order at most iv. ...##
###
Author index

2001
*
Linear Algebra and its Applications
*

Neogy

doi:10.1016/s0024-3795(01)00497-9
fatcat:6dtjvrfvqrgx3a4k6z6yykv2kq
*and*A.K. Das, More on positive subde®nite*matrices**and*the*linear*complementarity problem 338 (2001) 275 Nakazato, H.*and*P. ... . 338 (2001) 53 Wiesner, E.B., Backward minimal points for bounded*linear*operators on ®nite-dimen- sional vector*spaces*338 (2001) 251 Elsevier Science Inc. ...##
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On the eigenstructure of Toeplitz matrices

1984
*
IEEE Transactions on Acoustics Speech and Signal Processing
*

Thus, S(u) is the direct sum

doi:10.1109/tassp.1984.1164375
fatcat:aagz6qa6jvfqtijrxt5puy6zry
*of*the one-dimensional*space*spanned by w*and*the*space**of**Toeplitz**matrices*for which u is a singular vector. ... First we show that any reciprocal or anti-reciprocal n vector is an eigenvector for at least an [(n + 1)/2] -dimensional*linear**space**of*real symmetric n X n*Toeplitz**matrices*. ... The*matrices*D*and*E are, in general, complex*and*we assume that they are not*nilpotent*to avoid trivial cases. I. ...
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