A note on bordered isoclinal matrices

Journal of Computational North-Holland

and Applied

Mathematics

375

17 (1987) 375-377

A note on bordered isoclinal matrices M. MENEGUETTE Department Received

of Mathematics,

University of Strathclyde,

Livingstone

Tower, Glasgow Cl I XH, United Kingdom

22 July 1986

Abstract: An observation is made which links a class of bordered isoclinal matrices and companion a simple proof of a known result on the uniform boundedness of the inverses of that class.

matrices

In this note we consider form:

matrices

bordered

isoclinal

matrices

or marginally

Toeplitz

providing

of the

1

1 I 1 I I I I 1 I _----_----------_-+--_--_----_--_----_--_-__-_ 1 ffo a1 . ** ffk-l I

1

A,, =

ff0

ak-2

/

.. .

II I I 1

a0

ak-1

1 1

nxn

=

I kxk bn-kxk

9 kxn-k

Tn-kxn-k Toeplitz

1 ’

This class of matrix appears in the discretisation of ordinary differential equations and it has been shown that convergence depends upon the behaviour of its inverse ([4] but see also [6]). A thorough study of more general isoclinal matrices in the context of discretisation methods for more general operator equations was presented by McKee [3] who proved among other results: Theorem. There exist a constant M such that rnax(1a!,:‘)1,

A;l:=(aj,-‘)))

GM

if and only if the polynomial p( z) = zk + Ct:A LY/Z’ satisfies the root condition i.e., the zeros zi of p(z) are such that ) zj 1 < 1 with those equal to one being simple. Such p(z) is called a simple von Neumann polynomial. 0377-0427/87/$3.50

0 1987, Elsevier Science Publishers

B.V. (North-Holland)

376

M. Meneguette

/ Bordered isoclinical matrices

A slightly generalised version of this result seems to have been used and proved for the first time in [l] in connection with multistep methods for integral equations of the first kind (but see also [5] in the context of cyclic Linear Multistep Methods. Powers of the companion matrix C for p(z) furnish a much simpler and straightforward proof of the theorem above. Proof. For simplicity,

C= c3=

we consider,

[-to_lal]

thenC2=[

I

ao5+

matrix

C be in the form

iolI a[Tao]5

2 %-a0

*(yOal

a0 -aoa;

and in general

k = 2. Let the companion

%bo-

4)

1

with a(j) = _ “rajj-l)

_ aoajj-2)

for i = 1, 2;’ j >, 2 and C1 = C. On the other hand, from direct calculations 1

I 1

0 1’ --------;-----_------_--_----(1) a;) I 1 01 A-’ 00

=

(2) a1

a$2) 1 (1) ' a2

I

.

1

. . .

. . .

Toeplitz

Notice that A;l is also bordered isoclinal with the 2 first columns (k in general) generated by the last row of CJ (order k for the general case) and the entries for the Toeplitz block defined by the second column (k th in general). Therefore boundedness of the inverse above depends on the powers Cj alone. Now, Cj + constant matrix, as j --) cc, if and only if p(z) is a von Neumann polynomial; the theorem is then readily true. The proof for general k uses exactly the same argument. As a final comment, we point out that this proof gives rise to a natural link, in the context of discretization of ODE’s, between the usual definition of zero stability as in [2] and that given in t41.

References [l] P.A.W. Holyhead, S. McKee and P.J. Taylor, Multistep methods for solving linear Volterra integral equations of the first kind, SIAM J. Numer. Anal. 12 (1975) 698-711.

M. Meneguette [2] [3] [4] [5] [6]

/ Bordered isoclinical matrices

377

J.D. Lambert, Computational Methods in Ordinary Dijf erential Equations (Wiley, London, 1973). S. McKee, Discretization methods and block isoclinal matrices, IMA J. Numer. Anal. 3 (1983) 467-491. S. McKee and N. Pitcher, On the convergence of advanced linear multistep methods, BIT 19 (1979) 476-481. H.J. Stetter, Analysis of discretization methods for ordinary differential equations, 1973. P. Albrecht, Explicit, optimal stability functionals and their application to cyclic discretization methods, Computing 19 (1978) 233-249.