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Linear spaces of Toeplitz and nilpotent matrices

Richard A Brualdi, Keith L Chavey
1993 Journal of combinatorial theory. Series A  
We also give elementary proofs 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.  ... 
doi:10.1016/0097-3165(93)90025-4 fatcat:c442rshjbrbgbdifzqatryol6y

Report on the Dublin matrix theory conference, March 1984

N.B. Backhouse, A.G. Fellouris
1985 Linear Algebra and its Applications  
Undoubtedly, the most extensively studied infinite 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.  ... 
doi:10.1016/0024-3795(85)90221-6 fatcat:jqdxntdfzndkboxoyggwexq764

Isometries of the Toeplitz Matrix Algebra [article]

Douglas Farenick, Mitja Mastnak, Alexey I. Popov
2015 arXiv   pre-print
We study the structure 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) ].  ... 
arXiv:1502.01573v1 fatcat:66xti5ng7ja47cajgfhg7gp3k4

Products of commuting nilpotent operators

Damjana Kokol Bukovsek, Tomaz Kosir, Nika Novak, Polona Oblak
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.  ... 
doi:10.13001/1081-3810.1199 fatcat:5dgdzfnhqbf3zgoaziynzkr2zi

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

Dan Kucerovsky, Kaveh Mousavand, Aydin Sarraf, Nikos Katzourakis
2016 Cogent Mathematics  
We also give some results regarding circulant 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  ... 
doi:10.1080/23311835.2016.1154705 fatcat:4fm4d3vlr5hyri7kre3sxuhkbe

Patterned linear systems: Rings, chains, and trees

Sarah C. Hamilton, Mireille E. Broucke
2010 49th IEEE Conference on Decision and Control (CDC)  
In this second paper we study canonical patterns: rings, chains, 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.  ... 
doi:10.1109/cdc.2010.5717515 dblp:conf/cdc/HamiltonB10 fatcat:xsequz77krd7hmsls5xd7isopy

Efficient matrix preconditioners for black box linear algebra

Li Chen, Wayne Eberly, Erich Kaltofen, B. David Saunders, William J. Turner, Gilles Villard
2002 Linear Algebra and its Applications  
We present new conditioners, including conditioners to preserve low displacement rank for 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.  ... 
doi:10.1016/s0024-3795(01)00472-4 fatcat:4uc75rjqerborfo2bito2yn7jy

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?

D.Steven Mackey, Niloufer Mackey, Srdjan Petrovic
1999 Linear Algebra and its Applications  
The proof is constructive, 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  ... 
doi:10.1016/s0024-3795(99)00131-7 fatcat:uffivojszzewfcm6x6cj2y5vla

Representation spaces of the Jordan plane [article]

Natalia K. Iyudu
2012 arXiv   pre-print
We investigate relations between the properties 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  ... 
arXiv:1209.0746v1 fatcat:l6fo3q47zvam5kpvhn2gjbdanu

Irreducible components of the Jordan varieties [article]

N.Iyudu
2012 arXiv   pre-print
Along this line we establish an analogue 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  ... 
arXiv:0903.3820v3 fatcat:egsmt544wre53aixmskmob2zli

An operator not satisfying Lomonosov's hypothesis

D.W Hadwin, E.A Nordgren, Heydar Radjavi, Peter Rosenthal
1980 Journal of Functional Analysis  
An example is presented 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.  ... 
doi:10.1016/0022-1236(80)90073-7 fatcat:bm4nszvbe5ew7fi4oaszmzmrdm

Author index

2001 Linear Algebra and its Applications  
Neogy 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.  ... 
doi:10.1016/s0024-3795(01)00497-9 fatcat:6dtjvrfvqrgx3a4k6z6yykv2kq

On the eigenstructure of Toeplitz matrices

G. Cybenko
1984 IEEE Transactions on Acoustics Speech and Signal Processing  
Thus, S(u) is the direct sum 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.  ... 
doi:10.1109/tassp.1984.1164375 fatcat:aagz6qa6jvfqtijrxt5puy6zry
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