a matrix stability problem: problem 80-3
TRANSCRIPT
A Matrix Stability Problem: Problem 80-3Author(s): K. SourisseauSource: SIAM Review, Vol. 22, No. 1 (Jan., 1980), pp. 97-98Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2029880 .
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SIAM REVIEW iZ, 1980 Soctety, for lndusirmal and Appi;ed Mathematics Vol. 22, No. 1, January 1980 0036-1445/80/2201-0006$01 00/0
PROBLEMS AND SOLUTIONS
EDITED BY MURRAY S. KLAMKIN
COLLABORATING EDITORS: HENRY E. FETTIS (1885 California, Mountain View, CA 94041), YUDELL L. LUKE (University of Missouri, Kansas City, MO), CECIL C. ROUSSEAU (Memphis State University, TN), OTTO G. RUEHR (Michigan Technological UJniversity, Houghton, MI).
All problems and solutions should be sent, typewritten in duplicate, to Murray S. Klamkin, Department of Mathe,natics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. An asterisk placed beside a problem number indicates that the problem was submitted without solution. Proposers and solvers whose solutions are puiblished will receive 10 reprints of the corresponding problem section. Other solvers will receive just one reprint provided a self-addressed stamped envelope is enclosed. Proposers and solvers desiring acknowledgment of their contributions should include a self-addressed stamped postcard (no stamp necessary outside Canada). Solutions should be received by April 15, 1980.
PROBLEMS
A Determinant and an Identity
Problem 80-1 by A. V. BOYD (University of the Witwatersrand, Johannesburg, South Afica). (a) Prove that
det lArs = (1)n+1 (22n -2)B2n/(2n)!
where r,s =1,2, n,
ALrs=11(2r-2s+3)!, s---r+l, 0rs1 5s>r+1l,
and Bn is the Bernoulli number defined by
t c Bntn e'-1 n=0 ??!
(b) Prove that if n is odd, (n-1)/2 1-22m1 ( )2m )n+-2m )n+1-2mi
m= Y
Il 2m+12 h2 1B2m{(t+ hP~ -(t- h)+ m}
A Closed Form Integration
Problem 80-2* by A. VARMA (University of Notre Dame). The separable differential equation
du uu+lg+o}1/2 d = {u - uo + k log ku ? (0) = uO > ?, dx k +u u()uoO
arises in the solution of a diffusion-reaction problem. Can one obtain a closed form solution?
A Matrix Stability Problem
Problem 80-3* by K. SOURISSEAU (University of Minnesota) and M. F. DOHERTY (University of Massachusetts).
97
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98 PROBLEMS AND SOLUTIONS
Let A1 B1 C2 A2 B2
J- C,, A, Bn BNI1
CN AN
be a real square matrix of order 2N partitioned into N 2 x 2 blocks. The 2 x 2 blocks have the following structure:
-(Pn + Ln ) (Rn + Vn a)n -(Rn + Vn )e(n AH- Hn HtHn A,, =H,
--(Rn + Vn)Y -(-(Pn +LLn) (Rn + Vn)6n
Hn Hn Hn,,
Ln 1 Vn -lan -1 Vn -lgjn-1
n 0 Ln+1 VCn= lYnj
- Vn 16n-12
Hn ~H,, Hn,,
with an8n - yn3n. All the entries are nonzero positive constants with the exception of Pn and Rn, n = 1, 2,- , N which are nonnegative. The 2 x 2 blocks not shown contain only zeros and the eigenvalues of the nonzero blocks are:
A,,: A _- (Pn + Ln)< Hn
A 2 =-(P 'P+Ln (1 + an + din) < ?; H,,
B,,: Akl =A2 =L+> 0;
C,,: Al =0,
A 2 =V - 1(an-1 + An-1) > ?. Hn
What additional conditions must the elements of the matrix J satisfy in order that the eigenvalues of J have negative real parts?
The problem arose in considering the stability of a system of first order nonlinear ordinary differential equations describing the dynamics of a fractionation process.
Eigenvalues of a Tri-diagonal Matrix
Problem 80-4 by D. K. Ross (La Trobe University, Victoria, Australia). Prove that the real tri-diagonal matrix A = IIAiJii of order n has only real simple
eigenvalues if aiia, > 0 for j = i + 1.
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