A Matrix Stability Problem: Problem 80-3

A Matrix Stability Problem: Problem 80-3Author(s): K. SourisseauSource: SIAM Review, Vol. 23, No. 1 (Jan., 1981), pp. 107-112Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2029851 .

Accessed: 17/06/2014 03:52

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Review.

http://www.jstor.org

This content downloaded from 62.122.73.86 on Tue, 17 Jun 2014 03:52:00 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=siam

http://www.jstor.org/stable/2029851?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


PROBLEMS AND SOLUTIONS 107

Now substitute t = 2 and t = 1 in (3) and use B2n+Id)= B2n+1(1) =0 (n > 0).

Then (2) follows by subtracting the two equations. b) The right-hand side of (b) has the form

n Y, aih itn-j.

j=O

Clearly a, = 0 if i is odd. For i even, we find

a, = 1(13+1) m; (1-22m 1)( 2 )B2m,

which is 0 if j > 0 according to (2). Substituting h = 0, we find for the right-hand side

1-2- (t+h )n+1 -(th )n+1 n

lim - =t. h.o n+1 h

This proves (b). Also solved by C. GIVENS (Michigan Technological University), A. A. JAGERS

(Technische Hogeschool Twente, Enschede, the Netherlands), S. L. LEE (University of Alberta) and the proposer.

Additionally, LEE provides the generalization

det IAsI = (_.)k(2n+k+1)/2 det IB,,I,

where

A,s= 1/[2(r- s +k) +1]!, r,s=1,2, * *,n-k+l, k=0, 1, * *,n

(1/p! = O for p < O), and

Bij = #2(n+2-i-i)i is j = 1 2, . .. , k,

where

132m = (2m- 2)B2m/(2m)!.

His proof uses Sylvester's identity and induction.

A Matrix Stability Problem

Problem 80-3*, by K. SOURISSEAU (University of Minnesota) and M. F. DOHERTY (University of Massachusetts). Let

A1 B1 C2 A2 B2

Cn An Bn

* BN-1

CN AN



108 PROBLEMS AND SOLUTIONS

be a real square matrix of order 2N partitioned into N22 x 2 blocks. The 2 x 2 blocks have the following structure:

-(Pn + Ln) (Rn + Vn )a,n -(Rn + Vn)3,n

An Hn Hn Hn [- -(Rn + Vn )Yn -(Pn + Ln) (Rn + Vn,)8n L Hn Hn Hn

n+1 ?V,,an,I Vn]ln. 1I Hn ~~~H,, H,,

with a,,8,n = Yngn. 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

An: A1=- (P,+L,) < 0, H,,

A2 =- 4, n(1 + an + 8fn) < 0; H,,

Bn: A1=A2=-Ln+>0; H,,

C,,: A1=?,

A2= " (an-I + fn-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.

Solution by W. B. JORDAN (Scotia, NY). Note. The problem statement is incorrect. The second eigenvalue of An should be

A2 _ Pn +Ln Rn + Vn( + 8n) < 0 Hn Hn

For N = 1 the eigenvalues are real and negative, so the system is stable. We give stability conditions for N = 2, 3 and for a special case of N = 4, and we prove stability for all N if all stages are identical. We use the following notation. FN = AI2N - JN, SO IFNI = 0 is the characteristic equation.

[O 0] Q an n] 1 [an] a

hn= an + fn, rn = (Rn + Vn)/Hn,

pn = (Pn +Ln)/Hn, qn = Pn + rnhn,

Cn= VnLn+l/HHnHn+l, sOcn <rnPn+l, cnhn <pn+i(qn-Pn,),

E,, = Al-A,, =(A +Pn,)I +r,Q,,.




We find

En I=(A +Pn)(A+qn), En1 = I( I-A )

Q= n En Qn = = A + q A

In FN, divide the n + Ith and all subsequent column-pairs by Ln+l/Hn; restore the value of IFNI by multiplying the n + Ith and all subsequent row-pairs by the same factor, obtaining

E1 -I

-cQl Q E2 -I

FN=

-CN-2QN-2 EN-1 I -CN-1QN-1 ENJ

Let f1 = E1, f2 = E2f - c I Q 1, fn + 1 = En + Ifn-CnOnnf n - IQ 1 Multiply FN on the right by a matrix whose first column is I, fA, f2, , fN-I and diag is all I, so its determinant = 1. The product is

0 -I

O E2 -

0 -C2Q2 E3 -I

f fN 0 0 * -CN-1QN-1 EN

whose determinant is IfNI. Thus, fN is a 2 by 2 matrix whose determinant is the characteristic equation, and we have a recurrence formula for it.

If all stages are identical (En = E, cnOn = cQ) E and Q commute. The solution of the recurrence

fn + 1 = Efn -cQfn -1

which fits f1 and f2 is

fn = (2T)Y1[(2E + T)n+l- (E - T)n+l]

where T2 = IE2 - cQ. Let w = e 21ri/(n+1). Then, the factors of fn are 2E + T - w k(2E - T), for k = 1 to n + 1. Combine these factors in pairs, k = with k = n + 1 - j -j. n + 1 does not combine, and neither does 4(n + 1) when n is odd. Then,

Ifnl = 12TII(2E+ T)-( E-T)I HI['E+ T-'(d(E- T)][1E+ T-ct-'(E-T)]I.

The first factor cancels the 2 TI-1. The missing factor for n odd is l('E + T) + (2E - T)I =

IEl. The jth factor reduces to

IE-2cQ( +cosn- )I = (A+ p)2 [(A +q)2 -4ch cos n2 +jj




The eigenvalues are -A = p, q, q ?2Ich cos (jr/(n + 1)). The most positive (or least negative) is

-A =q-2v,Iccos >q-2lch>q-2-Ip(q-p) n + 1

= (lpVq-_p)2>O.

All eigenvalues are real and negative, and the system is stable. The transient response is likely to be sluggish, however, because of the multiple eigenvalue -p.

We return to the general case. If all Cn are 0, IFNI degenerates into the product of IEn 1, and is stable. It follows that the system is stable if the Cn are sufficiently small. Also, if A = -Pi the (1, 1) block of FN is r01Q, a multiple of the (2, 1) block. Since Q1 is singular IFNI= 0, SO -Pi is an eigenvalue for all N. The remaining characteristic polynomial is of degree 2N- 1. For the first few N,

f, = El, lfll = (A +pl)(A +ql).

For brevity, put (qlP2q2), etc., for (A +q1)(A +p2)(A +q2), etc., so Ifl (plql). Eigen- values are real and negative. The system is stable.

f2 = E2E1 - c1Ql = (I - c1Q1E'1E )E2E,

rciQI r2 Q2 \ y r I

E2E___ [ (q1P2) A +q2) 2

If21/IE1E21 = II + g1 Q1 + g2QIQ2|,

where g1 = -cl/(qlP2), g2 = clr2/(qlP2q2).

If M1, M2, M3 are 2 by 2 matrices, the identity

lk1Ml + k2M2 + k3M3l = (k1 - k2 - k3)k1lM1l + (k2 - k1 - k3)k2jM2I

+ (k3 - k - k2)k3lM31 + kIk2lMI +M21

+ k2k3|M2 +M31 + kIk3IM1 +M31

may be verified by direct expansion. Extension to more than three matrices is obvious. Since IQI =0 , this gives

If21/(plqlP2q2) = 1 -gi - g2 + glII + QlI + g21I + QlQ21.

For brevity, let ei, = aja1 + /3piy + yij3 + &i81 = trace QiQ1. Then

II+QI= 1+h1, II+QlQ2I= 1+e12,

If21/(plqlP2q2) = 1 + glhI + g2el2,

If21/(Pr) = (qlp2q2) - clhI(q2) + clr2e12

=A 3+alA 2+a2A +a3, where

al,= ql +p2 +q2, a2 = q1p2 +p2q2 +q2qI - c1h1,

a3 = qlp2q2 - clhlq2 + c1r2el2.

To test for stability we use the Lienard-Chipart modification of Routh's criterion. For a cubic with a, > 0 this requires

ala2> a3> 0.

But a 2qp ll 22q q l]>?




so stability for N = 2 requires only aIa2 > a3, or

(q, +P2)(q2 +P2)(q2 +qj) > cjhj(qj +P2) +clr2el2 Next,

f3 = E3f2 - c2Q2El

= E3(I + giQi + g2QiQ2 E- 2Q2) )E2E1

= E3(I + g1Q1 + g2Q0Q2 + g3Q2 +g4Q3Q2)E2E,,

where g3 = -c2/(q2p3), g4 = r3c2/(q2p3q3).

1f3/1E1E2E31 = 1 - g1 - g2 - g3 - g4 + glI + Q1l + g21I + Q1Q21 + g31I + Q21 + g411 + Q3Q21 + gig3Q1Q + Q21 + gig4IQ1 + Q3Q2 I

= 1 + gh+g2e12 + g3h2 + g4e23 + g9g3(h1h2 - e12) + g1g4d, where

d = 1Q1 + Q3Q21 = ai (y)3(z32 +8d352) -(,31(y)3t2 +83^Y2)

-y1(a3(32 + 13382) + o51(a3a2 + 133Y2), giving

Jf31/(p1) = (qiP2q2P3q3) - cjhi(q2p3q3) - C2h2(qIp2q3)

+ r2Cje12(p3q3) + r3C2e23(q1p2) + cIc2(hih2 - elAq3) - cIc2r3d

=A 5+aA 4+a2A 3+a3A 2+a4A +a5,

One can write out the formulas for the a's. Since a1 >0 and a2 >0, the Routh requirements are

a, a3 a5 0

a5>0, aja2>a3>0, and 0 a2 a4 0 >0. O a, a3 aS

0 1 a2 a4

We shall not attempt the general F4. However, if all Q, are alike, the En commute with Q and with each other, and

f2 = E2E1(I + gsQ),

f3 = E3E2E1(I + g6Q),

f4 = E4f3 - c3Qf2 = E4E3E2E1(I + g7Q),

where

g5 = gl + hg2 = -cl/(qlq2),

96 = gs + g3 + hg4= - cl/(qjq2) - C21(qAA

g7 = g6 - c3(1 + hgs)/(q3q4),

so 1f4j1/E1E2E3E4 = 1 + hg7

1f41/(PiP2P3P4) = [(q1q2) - hci][(q3q4) - hc3] - hc2(q1q4).




Let ul = qlq2- hcl >qiq2-p2(qi -pi) = qi(q2-P2)+P1P2>O and U2 = q3q4- hc3>O. Then,

f41/(P1P2P3P4) = [A2 + (q, + q2)A + u1][A2 + (q3 + q4)A + U2] - hC2(qlq4)

=A4+aA 3+a2A2+a3A +a4.

It is not difficult to show a,, a2, a3 > 0. The Routh requirements are then a4 > 0 and

a, a3 0

0< 1 a2 a4 =a3(aia2-a3)-aia4. 0 a, a3

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 = llaijll of order n has only real simple

eigenvalues if aijaji > 0 for j = i + 1. Editorial note. According to a number of our readers; R. ASKEY (University of

Wisconsin, Madison, Wisconsin), M. G. de BRUIN (Universiteit van Amsterdam, Amsterdam, the Netherlands), R. GORENFLO (Freie Universitat Berlin, Berlin, Federal Republic of Germany), J. G. SANDERSON (Los Alamos Scientific Laboratory, Los Alamos, New Mexico), Y. SOUN (Ford Aerospace and Communications Corporation, Houston, Texas) and H. WOLKOWICZ (University of Alberta, Edmonton, Alberta), this problem is well known, and quite a few references were included. In particular, J. Z. HEARON (National Institutes of Health, Bethesda, Maryland), enclosed a reprint of his paper, Irreducible continuant, Linear Alg. Appl., 3 (1970), pp. 125-127, which cites four solutions [1]-[4]. He obtains the solution as the following corollary of the following simple theorem on continuants and also cites three independent proofs of the corollary [6], [8], [9]:

THEOREM. Let Cbe any irreducible continuant. Then Cis diagonalizable if and only if the roots of C are distinct.

COROLLARY. The roots of a real, irreducible sign-symmetric continuant are real and distinct.

REFERENCES

[1] W. LEDERMANN AND G. E. H. REUTER, Spectraltheoryforthe differentialequations of simple birth and death processes, Phil. Trans. Royal Soc., 246 (1954), pp. 321-369.

[2] A. S. HOUSEHOLDER, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964. [3] S. YAMAMOTO, On the homogeneous birth and death processes with an absorbing barrier, Bull. Math.

Statist., 10 (1961), pp. 45-56. [4] J. H. SMITH, Problem 5578, Am. Math. Monthly, 75 (1968), p. 305. [5] J. Z. HEARON, Theorems on linear systems, Ann. N.Y. Acad. Sci., 108 (1963), pp. 36-68. [6] , The kinetics of linear systems with special reference to periodic reactions, Bull. Math. Biophys., 15

(1953), pp. 121-141. [7] , R?oots of an irreducible continuant, Linear Alg. Appl., 3 (1970), pp. 125-127. [8] S. PARTER, On the eigenvalues and eigenvectors of a class of matrices, J. Soc. Indust. Appl. Math., 8

(1960), 376-388. [9] J. S. MAYBEE, Matrices of class J2, J. Res. NBS, 71B (1967), p. 215.

[10] A. S. HOUSEHOLDER, Principles of Numerical Analysis, McGraw-Hill, New York, 1953. 111] E. T. BROWN, On the separation property of the roots of the secular equation, Am. J. Math.. 52 (1930),

pp. 843-850.

Also solved by CHICO PROBLEM GROUP (California State University, Chico, California), A. M. COHEN (U.W.I.S.T., Cardiff, U.K.), P. VAN DEN DRIESSCHE



Documents

A Matrix Stability Problem: Problem 80-3