Bounds for the Greatest Latent Root of a Positive Matrix

(1)

ON THE MEAN-IN DEPENDENCE OF TRANSLATES OF FUNCTIONS. 253

mean-independent. However, if we take a non-compact group and con-sider the space of all continuous functions with the topology of convergence uniform on every compact set, a novel situation arises in so far as the set of translates f(xa) is generally unbounded. I hope to return to this question in a later paper. Another problem of some interest would be to apply Theorem A to the study of the mean-invariant envelope of a specified set of translates of a given function, a question which appears not to have been discussed at all amidst the vast literature on linear envelopes of translates.

References.

1. R. E. Edwards, " On functions whose translates are independent " (to appear in the

Annalea de I'lnet. Fourier).

2. M. Krein and D. Milman, "On extreme points of regular convex sets", Studio Math., 9 (1940), 133-138.

3. H. Steinhaus, "Sur les distances des points dee• ensembles de mesure positive",

Fundamenta Math., 1 (1920), 93-104.

4. A. Weil, VintSgration dans les groupes topologiques et ses applications (Act. Sci. et Ind., No. 869, Paria, 1940).

Birkbeck College, London.

BOUNDS FOR THE GREATEST LATENT ROOT OF A POSITIVE MATRIX

A. OSTEOWSKI*f.

1. Let A ==• {a^) be an ?< X w-matrix with arbitrary non-zero u^.

Put /?„= £ U^ (n=\, . . . ,

n

) , ( I )

r = l H = m a x tiM, (2) it r - • min RM, (3) K = min 1^,1, (4) / / r—K \ . > a=

V{-R^)' &

* Received 10 October, 1951; road 15 November, 1951.

| This paper was prepared under contract of the National Bureau of Standards with the American University, Washington, D.C.

(2)

254 A. OSTROWSKI

Then the root co ivith the greatest modulus of the equation

\XE-A\ = 0 (6) satisfies the inequality

and, if all a^ are positive, the further inequality

(8)

2. The results (7) and (8) are an improvement of the corresponding inequalities given a year ago in an interesting note* by W. Ledermann.

Put

8=maxf'; (9)

RR U

then the inequalities of Ledermann are obtained from (7) and (8) on replacing a by y/h and ^ by < .

3. Since the modulus of <o is majored by the greatest fundamental root of the matrix (Ia^l), it is sufficient to consider the case in which all am are

positive. Then by a theorem of Perron and Frobenius, o> is positive and there exists a fundamental vector (xlt ..., xn) of A corresponding to cu, with positive xf:

n

o>a;,,= S altyxv (/x= 1, ..., n). (10) We shall prove a little more than the result stated, namely, assuming r<B,

minav

p K , < ~J : <<T. (11)

R—r-\-K max re.

We can assume, by permuting the rows and columns of A in a cogredient mannerf and multiplying all x, by a convenient constant, that

(12)

. * W. Ledermann, " Bounds for the greatest latent root of a positive matrix ", Journal

London Math. Soc., 25 (1950), 265-268.

f A cogredient transformation is the application of the same permutation to the rows and the columns.

(3)

BOUNDS FOR THE GREATEST LATENT ROOT OP A POSITIVE MATRIX. 256

Then we have from (10), (12) and (4), for any /x,

„!+ ( 2 %Axn = 0,^(1—xn)+RMxn,

In a similar manner it follows that, for any index A,

n - l

xKu> < S

4. We now specialize (13) and (14) by taking ^ and A such that

RM = R, Rx = r. (15)

Then it follows from (13) and (14), since x^^l, xx ^-xn, that

i.e. x ,(( i Z - K ) < w -K< — , (16)

and therefore „ < A/ \JT^-) — <*• (17)

We write now (1.3) and (14) for p = n, A = l , and obtain, since

, (18)

co </?—*+<«, (19)

that is (7) and (8).

"). On the other hand we have from (18), since the bound on the right-hand side in (IS) is less than R and Rn^r,

and solving this with respect to xn, we obtain

* » > T P ^ r R - (20)"

(4)

2 5 6 B 0 U N D 8 FOR THE GREATEST LATENT BOOT OF A POSITIVE MATRIX.

<v It m a y be r e m a r k e d finally t h a t t h e inequalities (7). (S) and (11) can be still further i m p r o v e d , b y i n t r o d u c i n g t h e expie.-sions

i<{ miii aw. K2 - nimalir. ( 2 1 )

Then in these inequalities we can replace a by

and K by K2.

University of Basel, Switzerland.

American University, Washington, D.C., U.S.A.

CORRIGENDA

ON A THEOREM DUE TO M. RIESZ

G. L. ISAACS*.

P. 289. In the conclusion of Theorem E, o(e-°") should be replaced by o(«* e~#_").

THE ASYMPTOTIC EXPANSION OF THE GENERALISED HYPERGEOMETRIC FUNCTION

E. M. P. 287, LEMMA, line 3, for Am read KAM;

line 6, for *•-•* read *-*-•*.

I am indebted to Dr. E. C. Bullard for drawing my attention to this error.

* This Journal, 26 (1951), 285-290. t This Journal, 10 (1935), 286-293.