Numer. Math. 24, 49--51 (1975) �9 by Springer-Verlag 1975

On Matrix Norms and Logarithmic Norms*

Emeric Deutsch

Received July 22, 1974

Summary. Let A be a complex n • n matrix, let r(A) denote the spectral radius of A and let ~t (A) denote the spectral abscissa of A. If v is a norm on C n, we denote by lub~ the matrix norm subordinate to v and by 7~ the logarithmic norm corresponding to v. New proofs are given for the following two relations: v(A)=influb~A and ~(A) = i n f 7~(A), where the infimums are taken over all ellipsoidal norms v on C'.

1~

Let C" denote the vector space of all n-tuples of complex numbers, let M,, denote the algebra of all complex n • n matrices, and let I denote the ident i ty matr ix in M,,. The spectral radius of a matr ix A EM~ will be denoted by r(A). If v is a norm on C n, let lub~ denote the matr ix norm on M~ which is subordinate to v [3]- I t is well-known tha t r(A) --<lub, A for every A c M n and for every norm on C n. Moreover, we have r(A) ----inf lub, A, where the inf imum is taken over all ellipsoidal norms v on C ~ [3]. In other words, given A EM~ and f l>r(A) , there exists an ellipsoidal norm v on C * such tha t

r(A) ~<lub, a <f t . (t)

Al though the proof of the existence of this norm v is a constructive one, the actual construct ion of v requires the tr iangularization of A.

We shall give a new construct ion of an ellipsoidal norm v, which, for a given A EM, and a given r > r(A), satisfies the inequalities (1). No tr iangular form of A is used in this construction.

If P is a positive definite matr ix in M,, we shall denote by II lie the norm on C * induced by the inner product (x, y)-+y* P x, where x, y EC ~ and y* denotes the conjugate transpose of the column vector y. Thus, II ll --(x*Px) for all xEC'. The matr ix norm subordinate to II I~, will be also denoted by II liP. I t can be easily seen tha t 11A ]b,=Vr(A*PAP -a) for all A EM~.

In our method we "scale d o w n " the given matr ix in order to ensure con- vergence (i.e. all eigenvalues inside the uni t circle) and then we make use of the following extension of P. Stein's theorem [71 due to O. Taussky [9] (see also [6]):

Let B E M n and assume that r ( B ) < t . Then, there exists a unique positive definite matrix P EM n such that P - - B* P B = I .

Our result, regarding the existence and the construct ion of a norm v satisfying (1) (for a given A EM, and a given fl > r (A)), is included in the following theorem.

�9 Research supported in part by National Science Foundation Grant GP-32834.

4 Numer. Math., Bd. 24

50 E. Deutsch

Theorem 1. Let A EM,, let f l>r (A) and let P be the positive definite matr ix in M,, such tha t

P--f l -2A* PA = I . (2) Then

r(A) <= HA [[~,=fl(t --(r(P))-l)t<fl. (3)

Proo/. The existence of a unique positive definite matr ix P satisfying (2) follows from the Stein-Taussky theorem quoted above and from the fact that r(fl-XA) < t. From (2) we obtain

A* PA P-a = r2 (I -- P-l). (4)

Since the left-hand side of (4) is similar to the positive definite matr ix (HAH-1) * �9 (HAH-1), where H is the positive definite square root of P, it follows that I - - p-1 has nonnegative eigenvalues (see also [9, Theorem 6]). If 21 is the largest eigen- value of P, then 1 - - ; t~ 1 is the largest eigenvalue of I - V-1 and so, taking spectral radii in both sides of (4), we obtain

]IA ]1~ = r (A* PA P-l) --_ f12( 1 -- (r (P))-I),

whence (3) follows at once.

Remark 1. I t can be easily seen that an explicit series expression of the matr ix P is given by (see also [6])

p = I +fl-2A* A +fl-~A**A2 + . . . .

Remark 2. Householder [3, P. t03] proves Stein's theorem by making use of the relation r ( A ) = i n f lub,(A), where the infimum is taken over all ellipsoidal norms v on C'.

2~

There exists a result, due to Pao [5], which is somewhat dual to Theorem 1" I t concerns the spectral abscissa of a matr ix and the logarithmic norm on Mn corresponding to a norm on C n. The spectral abscissa of a matrix A EM,, denoted

(A), is defined as the maximal real part of the" eigenvalues of A. For a norm v on C", the logarithmic norm y, corresponding to v is the real-valued mapping on M . defined by [I ]

y,(A) = l i m lub, (I +hA) -- t (A EM,). h~0 h

For example, the logarithmic norm corresponding to the Eudidean norm II II on C n is given by [t]

g(A) = ~ ( A + A * ) (A EM,). (5)

I t can be easily seen that the logarithmic norm corresponding to the norm It tle, denoted by gp, is given by

ge(A) = ~ ( A * + P A P -1) (A EM,). (6)

I t is known [t] that a(A) ~ y , ( A ) for every A EM~ and for every norm v on C ". I t is also known [5, 8] tha t ,t (A) = inf y, (A), where the infimum is taken over all ellipsoidal norms v on C n. Pao 's proof of this result [5] will be simplified.

Matrix Norms and Logarithmic Norms 51

Our proof is s imilar to the proof of Theorem t . W e "translate" the given m a t r i x in order to ensure s t ab i l i t y (i.e. all e igenvalues in the open left half-plane) and then we make use of L j a p u n o v ' s theorem [2, p. 270] (see also [4]):

Let B EM, and assume that oc(B)<0. Then, there exists a unique positive definite matrix P E M , such that P B + B* P = -- I.

Theorem 2. Let A EM n, let a > a (A) and let P be the posi t ive def ini te m a t r i x in M,, such t h a t

P (A - -a I ) + (A - -a I )* P = - - I . (7) Then

~. (A) <=gv(A) -= a --~ (r (P) ) -~ < a. (8)

Proo/. The existence of a unique posi t ive def ini te m a t r i x P sat isfying (7) follows from the L j a p u n o v theorem quo ted above and from the fact t ha t o~(A--aI) <0 . F r o m (7) we ob ta in

A* + PA P q = 2 a I - - p- l ,

and then, mak ing use of (6),

gp (A) ~--~ ~- 0~ ( 2 f f I - - p - l ) = a _{_10~ ( - - P - ' ) = a - - } (r (p))-x,

from which (8) follows a t once.

Remark 3. F r o m Theorem 2 i t follows t h a t equa l i t y prevai ls in the Corollary of E53.

References

1. Dahlquist, G. : Stabi l i ty and error bounds in the numerical integration of ordinary differential equations. Trans. Royal Inst i tute of Technology No. 130. Stockholm, Sweden, 1959

2. Franklin, J. N.: Matrix theory. Englewood Cliffs, New Jersey: Prentice-Hall t968 3, Householder, A. S.: The theory of matrices in numerical analysis. New York-

Toronto-London : t31aisdell 1964. 4. Ostrowski, A., Schneider, H. : Some theorems on the inertia of general matrices.

J. Math. Anal. Appl. 4, 72-84 (1962) 5. Pao, C. V. : Logarithmic derivates of a square matrix. Linear Algebra and Appl. 6,

159-164 (t973) 6. Redheffer, R. : Remarks on a paper of Taussky. J. Algebra 2, 42-47 (1965) 7. Stein, P.: Some general theorems on iterants. J. Res. Nat. Bur. Standards 48,

82-83 (1952) 8. Str6m, T.: On logarithmic norms. Report NA 69.06. Depar tment of Information

Processing, Computer Science, The Royal Ins t i tu te of Technology, Stockholm, Sweden, 1969

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Dr. Emeric Deutsch Polytechnic Ins t i tu te of New York Brooklyn, N.Y. t 120t U.S.A.

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