2
PROBLEMS AND SOLUTIONS 125 [3] W. K. HAYMAN, A generalisation of Stirling’s formula, J. Reine Angew. Math., 196 (1956), pp. 67-95. [4] P. BARRUCAND, Sur la somme des puissances des coefficients multinomiaux et les puissances succes- sives d’unefonction de Bessel, C. R. Acad. Sci. Paris S6r. Math., 253 (1964), pp. 5318-5320. [5] ., Problem 87-2", A multinomial summation, SIAM Rev., 30 (1988), pp. 128-130. Commuting Matrix Exponentials Problem 88-1", by DENNIS S. BERNSTEIN (Harris Corporation, Melbourne, Florida). In feedback control theory for sampled-data systems, the equivalent discrete-time dynamics matrix is given by e a, where h is the sample interval and A is the dynamics matrix for the original continuous-time system. When A is perturbed by A o (possibly due to some modeling uncertainty), then it is necessary to consider e <+). For robust control system design it thus may be of interest to know when e +) can be decomposed into a nominal part involving A and a perturbed part involving Ao. Analogous questions arise in the study of bilinear systems of the form c Ax + uBx, where u is a scalar control. In this case the Lie group generated by A and B plays a central role. Again, it is of interest to know how e A+ is related to e A and e , the principal result being the Baker-Campbell-Hausdorff formula. Of course, it is well known that when (1) AB=BA, where A, B are real n x n matrices, then both (2) eAe’=e’e A and (3) eAe=e A+ hold. It is less well known that the converse is not true. Specifically, examples are given in [1] showing that (2) may hold while (1) and (3) are violated, and that (3) may hold while (1) is violated. Interestingly, it is stated without proof in [1] that (3) implies (2). Prove this claim or find a counterexample. A copy of[l] is available from the proposer. REFERENCE M. FRICHET, Les solutions non commutables de l’bquation matricielle eXe y ex+r, Rend. Circ. Mat. Palermo (2), (1952), pp. 11-27. Solution by EDGAR M. E. WERMUTH (Zentralinstitut for Angewandte Mathematik, Jiilich, Federal Republic of Germany). The claim is false. An explicit counterexample is given by 0 0 0 0 0 0 0 A= 0 0 0 B= 0 0 0 0 0 a 0 0 0 0 0 b 0 0 0 Downloaded 12/06/14 to 216.165.95.79. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

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Page 1: Commuting Matrix Exponentials (Dennis S. Bernstein)

PROBLEMS AND SOLUTIONS 125

[3] W. K. HAYMAN, A generalisation of Stirling’s formula, J. Reine Angew. Math., 196 (1956), pp.67-95.

[4] P. BARRUCAND, Sur la somme des puissances des coefficients multinomiaux et les puissances succes-sives d’unefonction de Bessel, C. R. Acad. Sci. Paris S6r. Math., 253 (1964), pp. 5318-5320.

[5] ., Problem 87-2", A multinomial summation, SIAM Rev., 30 (1988), pp. 128-130.

Commuting Matrix Exponentials

Problem 88-1", by DENNIS S. BERNSTEIN (Harris Corporation, Melbourne, Florida).In feedback control theory for sampled-data systems, the equivalent discrete-time

dynamics matrix is given by ea, where h is the sample interval and A is the dynamicsmatrix for the original continuous-time system. When A is perturbed by Ao (possiblydue to some modeling uncertainty), then it is necessary to consider e<+). For robustcontrol system design it thus may be of interest to know when e+) can bedecomposed into a nominal part involving A and a perturbed part involving Ao.Analogous questions arise in the study of bilinear systems of the form

c Ax+ uBx,

where u is a scalar control. In this case the Lie group generated by A and B plays acentral role. Again, it is of interest to know how eA+ is related to eA and e, theprincipal result being the Baker-Campbell-Hausdorff formula. Of course, it is wellknown that when

(1) AB=BA,

where A, B are real n x n matrices, then both

(2) eAe’=e’eA

and

(3) eAe=eA+

hold. It is less well known that the converse is not true. Specifically, examples aregiven in [1] showing that (2) may hold while (1) and (3) are violated, and that (3)may hold while (1) is violated. Interestingly, it is stated without proof in [1] that(3) implies (2). Prove this claim or find a counterexample. A copy of[l] is availablefrom the proposer.

REFERENCE

M. FRICHET, Les solutions non commutables de l’bquation matricielle eXey ex+r, Rend. Circ. Mat.Palermo (2), (1952), pp. 11-27.

Solution by EDGAR M. E. WERMUTH (Zentralinstitut for Angewandte Mathematik,Jiilich, Federal Republic of Germany).The claim is false. An explicit counterexample is given by

0 0 0 0 0 0 0

A= 0 0 0 B= 0 0 00 0 a 0 0 00 0 b 0 0 0

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Page 2: Commuting Matrix Exponentials (Dennis S. Bernstein)

126 PROBLEMS AND SOLUTIONS

where z=a+ib is a solution of ez-z=l, e.g., a=2.088843 and b=7.461489 ... We get

cAcB cA+B0 0 0 a+l -b\

i 0 eeA=ilbal0 a+l b 0 a+l0 b a+ 0 b a+ 1/

Moreover, we can prove the following.THEOREM. IfA andB are square matrices with elements, then eAe eaeA ifand

only ifAB BA.

Also solved by J. BRENNER (Palo Alto, California), F. S. LEITE (Universidade deCoimbra, Portugal), and J. F. QUEER6 (Universidade de Coimbra, Portugal).

Editorial note. Solvers pointed out several relevant papers, including a correctionand counterexample published by Fr6chet in [2]. [C.C.R.]

REFERENCES

M. FRICHET, Les solutions non commutables de l’bquation matricielle eXer ex+ r, Rend. Circ. Mat.Palermo (2), (1952), pp. 11-27.

[2] ,Rectification, Rend. Circ. Mat. Palermo (2), (1953), pp. 71-72.[3] W. GIVENS, Review of Les solutions non commutables de liquation matricielle eXer ex+r,

M. FRCHET, Math. Reviews, 14 (1953), p. 237.[4] C. W. HUFF, On pairs of matrices (of order two) A, B satisfying the condition eAe= eA+B# eBeA,

Rend. Circ. Mat. Palermo (2), (1953), pp. 326-330.[5] A. G. KAKAR, Non-commuting solutions of the matrix equation exp (X + Y)= exp Xexp Y, Rend.

Circ. Mat. Palermo (2), (1953), pp. 331-345.

Zero of Least Modulus

Problem 88-3", by M. L. GLASSER (Clarkson University).The zeros of F(z) =- J(z) + J2(z) are of interest with respect to the properties of

water waves on a sloping beach [1]. One zero of F(z) is approximately given byz 2.977 + 1.266i. Does this correspond to the zero of least modulus?

REFERENCE

G. CARRIER AND H. GREENSPAN, Water waves offinite amplitude on a sloping beach, J. Fluid Mech.,4 (1958), pp. 97-110.

Solution by. MICHAEL RENARDY (Virginia Polytechnic Institute).The answer is yes. To show this, we simply need to evaluate

fcF’(z)_,2r’ F(z) az,

where C is a circle about the origin of radius r. This expression gives the number ofzeros of F inside the circle. Approximate values for the integral can easily be found

A proof and other results are given in an article entitled "Two results on matrix exponentials,"submitted to Linear Algebra Appl.

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