On matricial norms subordinate to vectorial norms

Math. Z. 122, 142-150 (1971) �9 by Springer-Verlag 1971

On Matric ia l N o r m s Subord inate to Vectorial N o r m s

EMERIC DEUTSCH

Introduction

Let C" denote the vector space of all n-tuples of complex numbers and let Rk+ denote the set of all k-tuples of nonnegative reals partially ordered componentwise. A vectorial norm of order k on C" is a mapping p: C"---~Rk+ such that

p(c~x)=]~lp(x) V ~ C , VxEC"; (V.1)

p ( x + y ) < p ( x ) + p ( y ) Vx, y~C"; (V.2)

p(x)~:O if x ~ 0 . (V.3)

Vectorial norms have been introduced by Kantorovitch [8]. Recently they have been studied by Robert [12, 13, 14], Stoer [17] and Deutsch [3].

We will denote by pa(X) . . . . . Pk(X) the components of p(x). It is clear that each of the mappings x ~ pj(x) (xe C") is a pseudonorm on C". To every vectorial norm we associate the following subspaces of C":

Kj(p)={x~C": pj(x)=0} ( j = l . . . . ,k),

Wj(p) = (~ Kh(P) (j = 1 . . . . , k). h:l: j

Properties of these subspaces can be found in [3] and [13]. A vectorial norm p: C"--,Rk+ is said to be regular if Wl(p)@.-. | Wk(P)= C".

Vectorial norms can be generated by the following procedure: let v be a vector norm on C", and let El, ..., Ek be the projections associated with a direct-sum decomposition C" = X 1 @ ... | X k of C". Then (see [5]), the mapping p: C"--* Rk+ defined by

x) ..... v(Ekx))

is a vectorial norm of order k on C", called the vectorial norm induced by the direct-sum decomposition {X 1 . . . . . Xk} and the vector norm v. It is known [-3] that a vectorial norm can be generated in this manner if and only if it is regular.

If p: C"~Rk+ is a vectorial norm and G is a non-singular n z n complex matrix, then the mapping

PG: C" ~ Rk+

p~(x)=p(G x) (xeC")

is also a vectorial norm, called the G-transform ofp [-3].

On Matricial Norms Subordinate to Vectorial Norms 143

Let M, denote the algebra of all n x n complex matrices and let M~- denote the set of all k x k nonnegative matrices partially ordered componentwise. A matricial norm of order k on M, [2] is a mapping #: M , - . M + such that

/~(~ A)= lal p(A) V a ~ C , V A e M , ; (M.1)

#(A+B)<t~(A)+I~(B) VA, B6M,; (M.2)

/z(AB)<=#(A)t~(B) VA, B6M,; (M.3)

#(A)+O if A + O . (M.4)

The concept of matricial norm has been mentioned by Wielandt [18] as a suggestion. They have been studied by Bauer [1], Robert [14], and Deutsch [2]. Particular mappings satisfying axioms (M.I-M.4) have been studied by Ostrowski [11], Nirschl [10], Robert [13], and Wielandt [18].

We will denote by pi~(A) the (i,j)-entry of #(A) (i,j= 1 . . . . . k). Obviously, a matricial norm of order k on M, is a vectorial norm of order k 2 on C "2 and therefore the concept of regularity makes sense also for matricial norms.

Matricial norms can be generated by the following procedure: let ~ be a matrix norm on M, [7], and let E~ . . . . . E k be the projections associated with a direct-sum decomposition C ' = X 1 | 1 7 4 k of C". Then (see [2]), the mapping/~: M,--+ Mk + defined by

12(A) = (O(EiAEj))i,j=I ..... k (A ~Mn)

is a matricial norm of order k on M,, called the matricial norm induced by the direct-sum decomposition {X 1 . . . . , Xk} and the matrix norm ~b.

If ~u: M, ~ Mk + is a matricial norm and G is a non-singular n x n complex matrix, then the mapp ing

#G(A)=I~(GAG -1) (AeM,)

is also a matricial norm, called the G-transform o f# [2].

Matricial Norm Subordinate to a Vectorial Norm

Let p be a regular vectorial norm of order k on C". It is known [13] that the mapping

lubv: M , ~ M~-

lubp A = (mij(A))i, j= 1 ..... ~ (A E M,), where

10"

pi(Ax) mu(A)= S u p -

~wj(vl pj(x) x 4 : 0

144 E. Deutsch:

is a regular matricial norm of order k on M,. We will call luPv the matricial norm subordinate to p. Clearly, this is a generalization of the concept of matrix norm subordinate to a vector norm [4],

Example 1. Consider the vectorial norm

p: C3--~RE+

p(~l , ~2,5~3) = (l~l], 1~2l "Jr ]~31) ((C(1, ~2, ~3)e C3).

We have Wl(p)={(a, 0,0): c~C} and Wz(p)={(0, fl, y): p ,?eC} and thus p is regular. If A = (aij)~,j= 1, 2, 3 sM3, one finds

luppA= [ [a111 Max {la121, la131} ] \laEl[+la311 Max{la22]+[a321, la231+laa3!} ]. (1)

It is known [13] that for a given regular vectorial norm p: C"~Rk+ and a given matrix A~M,, the matrix luppA is the least element of the set

{B~M~-: p(Ax)<_<Bp(x) V xeC"}.

Proposition 1. Let p: C"--*Rk+ be a regular vectorial norm, and let G be a non-singular n x n matrix. 7hen, the matricial norm subordinate to the G-transform PG of p is the G-transform (lubp) G oflubp.

(In other words, lubvG = (lubp)6).

Proof. For convenience, we will denote q = PG. The regularity of p implies that q is also regular [3]. Let El, ..., E k be the projections associated with the direct-sum decomposition C"= I471(/) ) |174 Wk(p) and let F 1 .... , F k be the projections associated with the direct-sum decomposition

C n = W l(q) (~.. . G Wk(q).

It is known [3] that Wj(q)= G-1 W~(p) ( j= 1, ..., k) and then it is easy to see that Fj=G-1EjG ( j = l . . . . . k). Now, we have for all i , j= l .... ,k

qi(A x) (lubqA)ij = Sup

x~w,(q) qj(x) x:l-O

or, since x = Fjx for x~ Wj(q)and qa(z)= qa(Fj z) for all z e C" ( j= 1 . . . . , k) [13], we obtain successively

(lubq A)o = Sup q~ (F~ AFj x) ~ a - ' w,(p) qa(Fj x)

x*O

= Sup q~(G-1E~GAG-*E~Gx)= Sup PI(EiGAG-1EjY) ~W,(P)x.o qa(G-'EjOx) Y~WJ(oP) Pa(EjY)

=-(lubp(GAG-1))ij

whence lubpG A = lubp (GAG- 1) = (lubp) G (A).


Example 2_ Consider the vectorial norm p of Example 1. Then the matricial norm subordinate to p is given by (1). Let

G= 1 .

0

The G-transform p~ of p is given by

p~(cq, ~2, ~x3)= (3 [cq[, 12~1+%1 + IC~31).

According to Proposition 1, the matricial norm lubp~ subordinate to PG is given by lubp~ A = lubp (GAG- 1) (A ~ M,) i.e.

where

, �9 [ s n s12 1 UDp~ Zt = ~ ]

"$21 $22 /

s n = ! a n - 2 a , 2 ] , s12 = 3Max{la121, lal3[},

s2, = �89 a,1 +a21 --4a,2 -2a221 + laat -- 2 aazl),

$22 = Max {12 a12 + a22 [-I-1a321, [2 a13 + 17231-t-]a331}.

(2)

Starting with a vector norm v on C" and a direct-sum decomposition C"=X~| ... @Xk of C", we can generate two matricial norms on M~:

(i) consider the vectorial norm p on C" induced by v and the given direct- sum decomposition of C" and then take the matricial norm lubp subordinate to p;

(ii) consider the matrix norm lub~ subordinate to v and then take the matricial norm # induced by lub v and the given direct-sum decomposition of C".

The following proposition gives the relation between these two matricial norms.

Proposition 2. Let El , . . . , E k be the projections associated with the direct- sum decomposition C"=XI@ ... @ X k of C", let v: C"-*R+ be a vector norm on C", let p: C ~ --~R~+ be the vectorial norm induced by {X 1 . . . . . Xk} and v and let #: M,-~M~- be the matricial norm induced by {X1, ..., Xk} and lubv. Then

(i) p(AB)<(lubpA) p(B) VA, B c M . ;

(ii) lubpA < #(A) < (lubvA) I~(I.) V A E M. (I1. is the identity matrix in M . ) ;

(iii) /f lub~ Ej = 1 for all j = 1, . . . , k, then # = lubp.

146 E.Deutsch:

Proof. From the appropriate definitions we have for all i , j= 1, ..., k

v (E i AEj x) (lubpA)i ~ = Sup pi(Ax) - Sup , (3) FJd pj(x) x,oX~X" v(x)

(# (A))ij = lub~ (E i AEj) = Sup v (E i AEj x) ~c~ v(x) x 4 : 0

(4)

Hence, for a given he{1 . . . . , k} we have for all x e X h (x#:O), ze C", A, B ~ M ,

v(EiAEhx) v(EhBEjz) (lubp A)i h (# (B))h j > v (x) v (z)

Taking X=EhBEjz and choosing ze C" such that EhBEjz=t=O, we obtain

V (E i AE h BE. z) (lub vA)~h(#(B))hj>=- v(z)

which holds for all z t 0 . Hence

v(EiAEhBE j z) (# (AE h B))ij = Sup v (z) <= (lubv A)ih (# (B))hj.

z * O

Now

(t2(AB))i j= (#(AE1B +.. . + AEgB))ij< (#(AE 1B))ij +.. . + (p(AE kB))ij

< (lubp A)i 1 (# (B))I j + . . , + (lubp A)i k (# (B)) k j

i.e. #(AB)<=(lubvA ) #(B). This proves statement (i). The first inequality of (ii) follows from (3) and (4) while the second one follows from (i) by taking B = I n. To prove statement (iii), note that #(I,)=diag(lubvE~, . . . , lub vEk). Taking into account the assumption lubvEj= 1 ( j= 1 . . . . , k) we obtain #(I , )=I k and now from statement (ii) we obtain #(A)=lubpA for all AEM, .

Remark 1. The following example shows that the inequalities (i) and (ii) can be strict. Let C S = X I @ X 2 where

x~={(~,-2~,0):~eC}, X2={(0,/L~,):/L~c}

and consider the vector norm v: C 3--+R+ defined by

v%, ~2, ~3)=-1~11 + 1~21 + 1~31.

The vectorial norm induced by {X a, X2} and v is

p: C 3 - + R 2

P(~I, ~ %)= (3 ]~t], 12at +~2!+ 1~31).


It is known [4] that the matrix norm lub~ subordinate to v is given by

lub~A= Max {[au[+la2jl+la3fl } j=1,2,3

where A = (aij)EM 3, and thus the matricial norm p induced by {X~, X2} and lub~ is

p: M 3 ----~M~-

#(A)= ( tit t12 )

\t21 t22/' where

t u = 3 [au--2al2[, t12= 3Max{2 [a12[, [a131},

tzl=12a11+ael-4ala-2a22[+[a31-2a3z[,

t22 --=Max {2 [2a12 q- a22 ] +2 1a321, 12a13 -]-a23 [ -1-1a331}.

The matricial norm lubp subordinate to p is given by

lubpA=( su sl2)

\$21 $22 ! where sn, s12, s21 , SeE are given by (2).

Remark2. Part (iii) of Proposition 2 shows that under the assumption lub~ Ej= 1 ( j= 1,..., k) the following diagram is commutative

V > p

1 + lub, - ~ # = lubp.

This property allows us to determine in some cases the matricial norm subordinate to a regular vectorial norm.

Example 3. Let C" = X~ 0 " " �9 Xk be an orthogonal direct-sum decomposition of C", and let E 1 . . . . . E k be the associated projections. If tl IJ is the Euclidean norm on C", then the vectorial norm on C" induced by {X a . . . . . Xk} and II is

p: Cn ---~ Rk+

p(x)=(llE~ xrl, ..., IlEkxll) (x~C").

The matrix norm Ill Ill subordinate to II II is given by [4]

III / l l l --1/r(A~) (A~Mn)

where A* is the adjoint of A and r(B) denotes the spectral radius of the matrix B. Since {X~ . . . . , Xk} is an orthogonal direct-sum decomposition of C", we have E*=Ej and so E*Ej=E*=E~ ( j = l . . . . ,k). Then IIIEflll--1 for all j = 1, ..., k. Now, making use of part (iii) of Proposition 2, we find that the matricial norm subordinate to p is given by

lUbpA=(l/r(EjA* EiAEfi)i,j=l ..... k (A6M.).

148 E.Deutsch:

The following proposition generalizes a theorem of Householder [-6].

Proposition 3. Let p: C"---~Rk+ be a regular vectorial norm such that W~(p)LWj(p)for all i , j= l . . . . . k, i+j. Then

lubp~ A = (lubp A*) T

for all A~M,. (B denotes the transpose of B and pD is the dual of p.)

Proof. It is known [3] that Wj(p D) = Wj(p) (j--- 1 . . . . . k). Let El, . . . , E k denote the perpendicular projections associated with the orthogonal direct-sum decomposition C"-- Wl(p)O"" �9 Wk(P). We have

(lubpDA)ij= Sup P~(EiAEjx) ~W~o~ p~(x) (5)

But p~(EiAEjx)= Sup I(E,AEjx)*z!= Sup Ix*EjA*E, zl

~ w ~ p~(z) ~w,~p) p~(z) z:l-O z t O

p~ (x) pj(e~ A* ei z) (6)

< Sup -p~(x) (lubpA*)ji - z~w,(p) pi(z)

z :~ O

where we have made use of inequality tu* v[ <p~(u) pj(v)(u~ C", v~ Wj (p)) which follows from the definition of the dual vectorial norm. Now (5) and (6) give (lubp. A)i j < (lubp A'*)ji whence

lubp, A < (lubpA*) T. (7)

Replacing in this inequality p by pD and A by A* and taking into account that pDD=p [-13] we obtain

(lubp A*) T < lubpD A,

which together with (7) proves our proposition.

The next proposition shows that if p and q are regular vectorial norms of order k on C" and lubpBNlubqB for all matrices B of rank at most one, then lubpA = lubqA for all AeM,,. This generalizes theorems of Schneider and Strang 1-15] and Ljubi6 [9] (see also Stoer [-16]). First we prove a lemma.

Lemma. Let p: C"-+ Rk+ be a regular vectorial norm. Then

lubp (x y*) = p (x) (pD (y))T for all x, y~ C".

Proof. We have for all i,j= 1,..., k

(lubp(xy*))ij= Sup pi(xy* z) - Sup lY* zl pi(x) =pi(x) p~(y). zsWAp)z,O p j ( z ) ZEzWJ~op) p j ( z )


Proposition 4. Let p, q: C"--* Rk+ be regular vectorial norms. The following statements are equivalent:

(i) lub v (x y*) < lubq (x y*) V x, y ~ C";

(ii) there exists a positive constant ~ such that p(x)=Tq(x) V xE C";

(iii) lubvA=lubqA V A~M,,; (iv) lubp A < lubq A V A ~ M,.

Proof The implications ( i i )~ (i i i)~ (iv)=> (i) are obvious. We prove that (i) implies (ii). By the preceding lemma, statement (i) can be written

p(x) (pn(y))r<q(x)(qD(y))r VX, y~ C". (8)

Let j e {1, ..., k} and let x e Wj(q). Then qi(x)= 0 for all i+ j and then from (8) we have pi(x)p~ (y)= 0 for all i+ j, for all h = 1 . . . . . k, and for all y e C". Hence p i (x)=0 for all i + j and therefore xe W~(p). Thus Wj(q)_~ Wj(p). Similarly, we can show that Kj(qD)c_Kj(pD). Now (see [13])

Wj(p) = (Kj(pD)) • ~ (Kj(qD)) • = Wj(q)

which together with Wj(q)~ W~(p)gives Wi(p)= Wj(q)for all j = 1 . . . . , k. Now, from (8) we have

pi(x) p~ (y) <= qi(x) qy. (y) (9)

for all x, y~ C" and all i , j= 1, ..., k. In particular, for all i= l . . . . . k and for all x, yE W/(p)= Wi(q) we have

Pi (x) p~ (y) < qi (x) q~ (y).

Since the restriction of p~ to W~(p) is the dual of the restriction of p~ to W~(p) [13], it follows [9] that there exists a constant c~ i> 0 such that p~(x)= 7i qi(x) for all x~ W~(p)= W~(q). Now from (9) we obtain

ei qi(x) ~]-1 qD. (y) < qi (x) q~ (y)

for all i , j = l , . . . ,k and for all x,y~Wi(p)=Wi(q). Hence a i<~ ~ for all i,j-- 1, ..., k and this implies ~i =~j ( i , j= 1 . . . . . k). Denoting ~ = ~1 . . . . . ~k we have pi(x)=~qi(x) for all x~Wi(p)= Wi(q) (i= 1 . . . . , k). If x~ C", we can write x = x l +.. . + x k with xj.~ Wj(p) = Wj(q) ( j= 1 . . . . . k) and then

pi(x) = pj(x2) = ~ q j(xj) = ~ q j(x),

where we have made use of the fact that pj(x)=pj(xj) and qj(x)=qj(x:) ( j= 1, ..., k) (see [13]).

Remark 3. From Proposition 4 it follows that if #1 and 1[12 are two distinct matricial norms on M, subordinate respectively to two vectorial norms of order k on C", then they are not comparable in the set of all matricial norms of order k on M, partially ordered in the obvious fashion.

150 E. Deutsch: On Matricial Norms Subordinate to Vectorial Norms

References

1. Bauer, F. L.: Theory of norms. Technical Report No. CS 75, Computer Science Department, Stanford University, 1967.

2. Deutsch, E.: Matricial norms. Numerische Math. 16, 73-84 (1970). 3. - On vectorial norms and pseudonorms. Proc. Amer. Math. Soc. 28, 18-24 (1971). 4. Faddeev, D. K., Faddeeva, V. N.: Computational methods of linear algebra. San Francisco-

London: Freeman and Co. 1963. 5. Fiedler, M., Ptak, V.: Generalized norms of matrices and the location of the spectrum. Czechosl.

Math. J. 12, 558-571 (1962). 6. Householder, A. S.: The approximate solution of matrix problems. J. Assoc. Comput. Machin.

5, 205-243 (1958). 7. - The theory of matrices in numerical analysis. New York-Toronto-London: Blaisdell 1964. 8. Kantorovitch, L. V.: The method of successive approximations for functional equations. Acta

Math. 71, 63-97 (1939). 9. Ljubi~, Ju. I.: On operator norms of matrices. Uspehi Mat. Nauk 18, No. 4 (112), 161-164

(1963) [in Russian~. 10. Nirschl, N. E.: Applications of norms to eigenvalue localization theorems and field of values

theorems in matrix theory. Doctoral dissertation, University of Wisconsin, 1963. 11. Ostrowski, A. M.: On some metrical properties of operator matrices and matrices partitioned

into blocks. J. Math. Anal. Appl. 2, 161-209 (1961). 12. Robert, F.: Normes vectorielles de vecteurs et de matrices. Reve Franq. Traitement Inform.

(Chiffres) 7, 261-299 (1964). 13. - Sur les normes vectorielles r6guli6res sur un espace vectoriel de dimension finie. C.R.

Acad. Sci. Paris 261, 5173-5176 (1965). 14. - Etude et utilisation de normes vectorielles en analyse num6rique lin6aire. Doctoral

dissertation, Universitb de Grenoble, 1968. 15. Schneider, H., Strang, W. G.: Comparison theorems for supremum norms. Numerische Math.

4, 15-20 (1962). 16. Stoer, J.: On the characterization of least upper bound norms in matrix space. Numerische

Math. 6, 302-314 (1964). 17. - Lower bounds of matrices. Numerische Math. 12, 146-158 (1968). 18. Wielandt, H.: Topics in the analytic theory of matrices. Lecture notes (prepared by R.R.

Meyer), Department of Mathematics, University of Wisconsin, 1967.

Dr. Emeric Deutsch Department of Mathematics Polytechnic Institute of Brooklyn Brooklyn, N.Y. 11201, U.S.A.

(Received September 29, 1970)

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On matricial norms subordinate to vectorial norms