Pairs of matrices with a non-zero commutator

Goddard, Linda; Schneider, Hans

doi:10.1017/s0305004100030632

Cited by 14 publications

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“…If A and B are quasi-commutative, then clearly B is a commutant of A and ωA. The general question of commutants was also considered by Goddard-Schneider [14]. One might observe that all the mathematicians mentioned in this paragraph were in Scotland in the early 1950's.…”

Section: Introductionmentioning

confidence: 88%

Potter, Wielandt, and Drazin on the Matrix Equation AB = ωBA: New Answers to Old Questions

Holtz

Mehrmann

Schneider

2004

The American Mathematical Monthly

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In this partly historical and partly research oriented note, we display a page of an unpublished mathematical diary of Helmut Wielandt's for 1951. There he gives a new proof of a theorem due to H. S. A. Potter on the matrix equation AB = ωBA, which is related to the q-binomial theorem, and asks some further questions, which we answer. We also describe results by M. P. Drazin and others on this equation.

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Section: Introductionmentioning

confidence: 88%

Potter, Wielandt, and Drazin on the Matrix Equation AB = ωBA: New Answers to Old Questions

Holtz

Mehrmann

Schneider

2004

The American Mathematical Monthly

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“…For, if p is an isomorphism merely of the subalgebra F[b] of 8(IF) into 8(F), and C = b p , then we can only conclude that JU,,(1F, /) = M^(F, /). Goldhaber (2) and Goldhaber and Whaples (3) have proved that if A and B are square matrices with coefficients in F such that there exists a non-singular matrix P for which N = A -PBP~l lies in the radical of F[A, PBP- 1 ] then \tl -A\ = \tl -B\. This result is generally known as Goldhaber's lemma.…”

Section: Corollary 2 If \P Belongs To the Characteristic Ideal Of Amentioning

confidence: 99%

Characteristic Polynomials

Schneider

1957

Can. j. math.

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Introduction. Let F be a field and let V be a finite dimensional vector space over F which is also a module over the ring F [a]. Here a may lie in any extension ring of F. We do not assume, as yet, that V is a faithful module, so that a need not be a linear transformation on V. It is known that by means of a decomposition of V into cyclic F[a]-modules we may obtain a definition of the characteristic polynomial of a on V which does not involve determinants. In this note we shall give another non-determinantal definition of the characteristic polynomial. Instead of considering a single module V, we shall accordingly study the set of all finite dimensional F[a]-modules and mappings of this set into monic polynomials with coefficients in F. Admittedly our procedure does not yield the theory of the elementary divisors of a, but it has certain advantages. First, all questions of uniqueness are settled immediately by the Jordan-Holder theorem. Secondly, it is possible to derive some classical results, usually proved using determinants, without excessive labour. To illustrate the use of our method we shall complete and generalise some results due to Goldhaber (2) and Osborne (5). Received March 1, 1956. I should like to express my gratitude to Dr. I. T. Adamson and Dr. M. P. Orazin for discussing with me the results of this paper. . . CHAI{ACTEIUSTIC POLYNOMIALS 67 THEOREM 4. Let a and b be linear transformations on tlte vector spaces V and W respectively. Let X be a vector space of dimension rover F. Then the characteristic polynomials Xa (V, t) and Xb (W, t) have a common factor of degree r if and only if there exist an F[a]-module Z contained in V and a homomorphism (}' of S(Z, V) onto 2(X), an F[b]-module Y contained in Wand a homomorphism T of S(Y, W) onto 2(X), such that n = au -b T lies in the radical of F[a u , bTl.Finally, we claim that some of the results of Goddard and Schneider (1) and other results on characteristic polynomials may be derived from Theorem 4.

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“…matrices which is given ill the n ext section. The only new res ult hore is the formula for a maLrix: satisfying (5), as the formula for H, m a tr ix s<tLisfying (1) was given by Goddard and Schneider [4], and the one for a matrix satisfyin g (3) was given by Lhe a,uthor in [6]. The three transformation maLri ces axe given below, however , in order to show tho rclaLions among them, but the proof is not give n in d eUtil as it appear s in the other paper s.…”

Section: A [P(i Px Ajq]= [P(ipxx)q]bmentioning

confidence: 99%

“…Schneider [4]1 discussed the relations bip beLween maLrices A and B, of orders 11. and m respeclively, which salisfy AX= XB (1 ) for some n X m matrix, X , of fH,nk 1'> 0.…”

Section: Introductionmentioning

confidence: 99%

“…) It was shown by Lhe auLhor in [6] that, given a malrix A saLisfying (3) for some maLrices B and X , Lhere exisLs a Illitll'ix P , \ 'hich Ctlll be con stru clcd from X , sllch thaL (4) so thaL pr of the rooLs of A are Lhe rools of B. Thus, if 1'<11. , or of B is r educcd, (4 ) gives fl, rcduclion formula for A. Naturally a maLrix satisfying (1 ) sfl,tisfi es (3) with p = 1.…”

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confidence: 99%

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Special types of partitioned matrices

Haynsworth¹

1961

J. RES. NATL. BUR. STAN. SECT. B. MATH. MATH. PHYS.

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This paper extends t he results of two previous papers on partition ed matrices. General r ed uction formulas are given for pa rtition ed matrices A of order np sa tisfying A (X X J p ) = (X X J p) B, \I' h ere B is a matrix of order r p and X X J v r epresen ts the direct producL of a n n X r matrix X of rank r with th e id en tity matrix of order p . These formu las are r elated to the formulas given in t he previous papers for pa r tition ed matrices satisfy ing A (J pX X ) = (J pX X ) B .

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Pairs of matrices with a non-zero commutator

Cited by 14 publications

References 4 publications

Potter, Wielandt, and Drazin on the Matrix Equation AB = ωBA: New Answers to Old Questions

Potter, Wielandt, and Drazin on the Matrix Equation AB = ωBA: New Answers to Old Questions

Characteristic Polynomials

Special types of partitioned matrices

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