Pairs of matrices with property 𝐿

Motzkin, T. S.; Taussky, Olga

doi:10.1090/s0002-9947-1952-0049855-8

Cited by 6 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Definition (Motzkin and Taussky, 1952). A pair of square matrices K and D have property P if for any suitable function/-e.g.,/analytic in any region containing the spectra of K and D-\,-[f(K, D)J = f(XiK, X/3), where the pairing of the eigenvalues \iK and XiD is independent of the function /•…”

Section: Mathematical Formulationmentioning

confidence: 99%

Interactions of Reaction and Diffusion in Open Systems

Othmer¹,

Scriven²

1969

Ind. Eng. Chem. Fund.

101

View full text Add to dashboard Cite

Section: Mathematical Formulationmentioning

confidence: 99%

Interactions of Reaction and Diffusion in Open Systems

Othmer¹,

Scriven²

1969

Ind. Eng. Chem. Fund.

101

View full text Add to dashboard Cite

“…The structure of the varieties of commuting pairs of matrices and of commuting pairs of nilpotent matrices is not yet well understood. It was proved by Motzkin and Taussky [13] (see also Guralnick [7]), that the variety of pairs of commuting matrices was irreducible. It was Guralnick [7] who showed that this is no longer the case for the variety of triples of commuting matrices (see also Guralnick and Sethuraman [8], Holbrook and Omladič [11], Omladič [14], Han [10],Šivic [16]).…”

Section: Introductionmentioning

confidence: 99%

The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix

Oblak

2008

Linear and Multilinear Algebra

View full text Add to dashboard Cite

Abstract. We study the set P(N B ) of all possible Jordan canonical forms of nilpotent matrices commuting with a given nilpotent matrix B. We describe P(N B ) in the special case when B has only one Jordan block and discuss some consequences. In the general case, we find the maximal possible index of nilpotency in the set of all nilpotent matrices commuting with a given nilpotent matrix. We consider several examples. Math. Subj. Class (2000): 15A04, 15A21, 15A27

show abstract

“…The first results on ordinary commuting varieties were the independent proofs by Motzkin-Taussky [23] and Gerstenhaber [12] that 𝐶 2 (𝔤𝔩 𝑛 ) is irreducible. Subsequently, this was extended to the Lie algebra of an arbitrary reductive algebraic group by Richardson [31] in characteristic zero, and by the first author [18] in positive characteristic (under mild hypotheses).…”

Section: Commuting Varietiesmentioning

confidence: 99%

Commuting varieties and cohomological complexity theory

Lévy

Ngo

Šivic

2022

Journal of London Math Soc

View full text Add to dashboard Cite

In this paper we determine, for all r$r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting r$r$‐tuples of nilpotent elements of frakturgln$\mathfrak {gl}_n$. Our main result is that in characteristic ≠2,3$\ne 2,3$, this nilpotent commuting variety has dimension (r+1)⌊n24⌋$(r+1)\lfloor \frac{n^2}{4}\rfloor$ for n⩾4$n\geqslant 4$, r⩾7$r\geqslant 7$. We use this to find the dimension of the (ordinary) r$r$th commuting varieties of frakturgln$\mathfrak {gl}_n$ and fraktursln$\mathfrak {sl}_n$ for the same range of values of r$r$ and n$n$. Our principal motivation is the connection between nilpotent commuting varieties and cohomological complexity of finite group schemes, which we exploit in the last section of the paper to obtain explicit values for complexities of a large family of modules over the r$r$th Frobenius kernel (prefixGLn)(r)$(\operatorname{GL}_n)_{(r)}$. These results indicate an inequality between the complexities of a rational G$G$‐module M$M$ when restricted to G(r)$G_{(r)}$ or to Gfalse(Fprfalse)$G(\mathbb {F}_{p^r})$; we subsequently establish this inequality for every simple algebraic group G$G$ defined over an algebraically closed field of good characteristic, significantly extending the main theorem of Lin and Nakano, Inventiones Mathematicae, 138 (1999), 85–101.

show abstract

Pairs of matrices with property 𝐿

Cited by 6 publications

References 7 publications

Interactions of Reaction and Diffusion in Open Systems

Interactions of Reaction and Diffusion in Open Systems

The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix

Commuting varieties and cohomological complexity theory

Contact Info

Product

Resources

About