• Corpus ID: 118301403

Potter, Wielandt, and Drazin on the matrix equation $AB=\omega BA$, with some new answers to old questions

@article{Holtz2005PotterWA,
  title={Potter, Wielandt, and Drazin on the matrix equation \$AB=\omega BA\$, with some new answers to old questions},
  author={Olga Holtz and Volker Mehrmann and Hans Schneider},
  journal={arXiv: Rings and Algebras},
  year={2005}
}
In this partly historical and partly research oriented note, we display a page of an unpublished mathematical diary of Helmut Wielandt's for 1951.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 $AB = \omega BA$, 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. 

References

SHOWING 1-10 OF 14 REFERENCES
A reduction for the matrix equation AB = ε BA
1. It is well known that, if two n × n matrices A, B commute, then there is a non-singular matrix P such that P −1 AP , P −2 BP are both triangular (i.e. have all their subdiagonal elements zero).
The q-binomial theorem and spectral symmetry
In various contexts, several mathematicians have discovered a binomial theorem of the following form: Let T1,T2 be complex matrices such that T2T1 = qT1T2. Then (T1 + T2)n = SIGMA(k = 0)n
Simultaneous similarity of matrices
Abstract In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT−1, TBT−1). Roughly
II. A memoir on the theory of matrices
  • A. Cayley
  • Engineering
    Philosophical Transactions of the Royal Society of London
  • 1858
The term matrix might be used in a more general sense, but in the present memoir I consider only square and rectangular matrices, and the term matrix used without qualification is to be understood as
Basic Hypergeometric Series
Foreword Preface 1. Basic hypergeometric series 2. Summation, transformation, and expansion formulas 3. Additional summation, transformation, and expansion formulas 4. Basic contour integrals 5.
PAIRS OF MATRICES WITH A NON-ZERO COMMUTATOR
l. This note takes its origin in a remark by Brauer(l) and Perfect(5): Let A be a square complex matrix of order 11, whose characteristic roots are al' ... , an' If Xl is a characteristic column
Quantum Groups
Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups
The Theory of Matrices
Volume 2: XI. Complex symmetric, skew-symmetric, and orthogonal matrices: 1. Some formulas for complex orthogonal and unitary matrices 2. Polar decomposition of a complex matrix 3. The normal form of
Symmetries and variation of spectra
An interesting class of matrices is shown to have the property that the spectrum of each of its elements is invariant under multiplication by p-th roots of unity. For this class and tor a class of
...
1
2
...