Comment on "Generators of matrix algebras in dimension 2 and 3"
CCOMMENT ON “GENERATORS OF MATRIX ALGEBRAS IN DIMENSION 2AND 3” ´ANGELA CAPEL AND YIFAN JIA
Abstract.
Theorem 7 in [AS09] states sufficient conditions to determine whether a pair generatesthe algebra of 3 × discussion Let M n ( K ) denote the set of all n × n matrices over a field K . Let S be a subset of M n ( K ) anddenote by S m the set of all products of the form A · · · A m , with A i ∈ S ∪ { I n } for all i = 1 , . . . , m ,where I n is the n × n identity matrix. We say that a generating set S has length k ∈ N ifspan (cid:110) S k (cid:111) = M n ( K ) , and span (cid:110) S k − (cid:111) (cid:40) span (cid:110) S k (cid:111) . The problem of finding bounds on the length of generating sets, and in particular generating pairs,has been thoroughly studied in the past decades. For arbitrary order n , the best known bound onthe length of any generating set is O ( n log n ) [Shi19], although it was conjectured years before thatthe optimal bound might be 2 n − n ≤ n underdifferent conditions on one of the generators [LR11, GLMˇS18], even though it is always possible tofind n matrices in M n ( K ) such that the words of length 2 in those matrices span the whole algebra[Ros12]. Finally, when the problem is reduced to the study of generic matrices, the bound can bearguably improved to O (log n ) [KˇS16].In [AS09], the problem of providing conditions under which a set of 2 × × A and B . Unless statedotherwise, we follow [AS09] for terminology and notations.Before recalling Theorem 7 of [AS09], we denote by M the algebra of 3 × A, B ] for the commutator oftwo matrices A and B and we define H ( M ) := tr[ M ] − tr (cid:2) M (cid:3) , where tr[ M ] denotes the trace of a matrix M . We can now state the aforementioned result, namelyTheorem 7 of [AS09]. Date : February 11, 2021.
Keywords:
Generator; Matrix; Algebra.
Primary 15A30; Secondary 47L05. a r X i v : . [ m a t h . R A ] F e b ´ANGELA CAPEL AND YIFAN JIA Theorem 7.
Let
A, B ∈ M . Then det (cid:0) I, A, A , B, B , AB, BA, [ A, [ A, B ]] , [ B, [ B, A ]] (cid:1) = 9 det[ A, B ] H ([ A, B ]) , (1) so if det[ A, B ] (cid:54) = 0 and H ([ A, B ]) (cid:54) = 0 , then { I, A, A , B, B , AB, BA, [ A, [ A, B ]] , [ B, [ B, A ]] } form a basis for M . Note that this result intends to provide a sufficient condition for a pair of matrices { A, B } to generate the full algebra M , as well as construct a basis for the algebra from words in suchmatrices. In particular, if the result was correct, it would yield the fact that for pairs { A, B } suchthat det[ A, B ] (cid:54) = 0 and H ([ A, B ]) (cid:54) = 0, the necessary length of words to generate the full M is3, improving thus for this subclass of pairs the bound of 2 n − (cid:100) log (3) (cid:101) = 4. Thus, Theorem 7 wouldalso provide an improvement to the latter result in dimension 3.To show that Equation (1) is false, and thus Theorem 7 does not hold in general, it is enough toconstruct a counterexample. For any δ > , ε >
0, consider the following pair of matrices: A = − δ , B = − ε − . Then, a short computation yields the following:det (cid:0)
I, A, A , B, B , AB, BA, [ A, [ A, B ]] , [ B, [ B, A ]] (cid:1) = − δ ε + 9 δ ε , A, B ] H ([ A, B ]) = 27 δ ε − δ ε . Clearly, we can conclude that, for A and B as above, the left- and right-hand sides of Equation (1)are different, although we appreciate that both vanish if, and only if, either ε or δ is zero. Thus,despite being different, both sides of the equation present a strong correlation. This is noticeablein Figure 1. Figure 1.
Test for 5000 randomly chosen 3 × OMMENT ON “GENERATORS OF MATRIX ALGEBRAS IN DIMENSION 2 AND 3” 3
In Figure 1, we generate 5000 random matrix pairs and compare their value when we insert thepair into the LHS and the RHS of Equation (1). The points do not lie on some line f ( x ) = ax + b ,especially not on f ( x ) = x . Therefore, we exclude the possibility that the incorrectness is causedby the improper calculation of coefficient 9 in the RHS of Equation (1). Moreover, if we restrictthe data to the first and the third quadrants close to zero, the best fitted line has slope a ≈ b ≈
0. For example, for the data in Figure 1, the program returns f ( x ) ≈ . x +0 . x ”.To conclude, we have shown that Theorem 7 in [AS09] is false. For a correct upper bound on thenecessary length of words on a pair to generate M , we refer the reader to [LNP06] for the generalcase and to [KˇS16] for the generic case.After the completion of this note, we realized that for Equation (1) to be true, there were afactor det[ A, B ] and a minus sign missing in the right hand side. Therefore, we present the correctform of Theorem 7 below.
Theorem 7 revised.
Let
A, B ∈ M . Then det (cid:0) I, A, A , B, B , AB, BA, [ A, [ A, B ]] , [ B, [ B, A ]] (cid:1) = − A, B ]) H ([ A, B ]) , so if det[ A, B ] (cid:54) = 0 and H ([ A, B ]) (cid:54) = 0 , then { I, A, A , B, B , AB, BA, [ A, [ A, B ]] , [ B, [ B, A ]] } form a basis for M . This yields the fact that whenever det[
A, B ] (cid:54) = 0 and H ([ A, B ]) (cid:54) = 0, the set composed of wordson matrices A, B , which was presented in the original paper [AS09], actually spans the whole matrixalgebra M . In particular, the necessary length of words to generate the full M is 3. As mentionedabove, this indeed improves the bound 2 n − (cid:100) log ( n ) (cid:101) provided in [KˇS16] for generic matrices in dimension n = 3. Acknowledgments.
We thank Michael Wolf for his comments and suggestions. This work has beenpartially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)under Germany’s Excellence Strategy EXC-2111 390814868. YJ acknowledges support from theTopMath Graduate Center of the TUM Graduate School and the TopMath Program of the EliteNetwork of Bavaria.
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Email address : [email protected] Department of Mathematics, Technische Universit¨at M¨unchen, 85748 Garching, Germany and Mu-nich Center for Quantum Science and Technology (MCQST), M¨unchen, Germany
Email address : [email protected]@tum.de