Internat. J. Math. & Math. Sci.
VOL. 19 NO. 4 (1996) 707-710
707
ON MATRIX CONVEXITY OF THE MOORE-PENROSE INVERSE
J.E. PE(ARI(
Faculty of Textil Technology
University of Zagreb
Zagreb, CROATIA
B. MOND
Department of Mathematics
La Trobe University
Bundoora, Victoria, 3083, AUSTRALIA
(Received December 12, 1994 and in revised form June 28, 1995)
ABSTRACT. Matrix convexity of the Moore-Penrose inverse was considered in the recent literature
Here we give some converse inequalities as well as further generalizations
KEY WORDS AND PHRASES: Matrix convexity, generalized inverse
1991 AMS SUBJECT CLASSIFICATION CODES: 15A45, 15A09.
1.
INTRODUCTION
Let A and B be two complex Hermitian positive definite matrices, and let 0 _< A _< 1 Then
A)B] -1 <_ AA -1 + (1
[AA + (1
A)B -1
(1 1)
where A _> B means that A B is a positive semi-definite matrix.
This result, e., matrix convexity of the inverse function is an old result that appears explicitly in the
papers 1,2,3,4,5] (see also the books [6, pp 554-555] and [7, pp. 469-471]).
The related matrix convexity of the Moore-Penrose (generalized) inverse, denoted by A /, was
considered in paper [8,9,10] The following was given in 10]"
Let A and B be two complex Hermitian positive semi-definite matrices of the same order. The
inequality
[AA + (1 A)B] + _< AA + + (1 A)B +
(1 2)
for every 0 < A < 1 holds if and only if
R(A)
R(B)
(1 3)
where R (A) is the range of A.
Two converses of (1.1) were obtained in 11 ]"
If A and B are complex Hermitian positive definite matrices and 0 < A < 1 is a real number, then
[AA + (1
A)B] -1 _> K(AA -1 + (1
A)B -1)
(1 4)
and
[AA + (1 A)B]-’
(AA -1 -I- (1
A)B -1) _>
where
g
min
(1+#,)2’
R=min
RA-’
v-
(1.5)
(1 6a, b)
_#,
and the # are the solutions of the equation
det(B #A)
0.
(1 7)
In this note, we give analogous converses for (1 2), as well as some related results
CONVERSES OF TIlE MATRIX CONVEXITY INEQUALITY
OF THE MOORE-PENROSE INVERSE
Let A and B be two complex Hermitian positive semi-definite matrices of the same order such that
(1 3) holds Let P be a unitary matrix such that A P diag (A1,0)P* where A1 is a diagonal positive
definite matrix When (1 3) holds, we have B P diag (B1,0)P* where B1 is positive definite
2.
708
E PEOARI(
B MOND AND
THEOREM 1. Let A and B be two complex Hermitian positive semi-definite matrices of the same
order such that (1 3) holds and let 0 _< A < 1 Then
[AA + (1
A)B] * >_ K(AA + + (1
A)B +)
(2 l)
where K is defined by (1 6a) and the #, are the positive solutions of the equation
det(B
AA)
(22)
0.
TItEOREM 2. Let A, B be defined as in Theorem
Then
[AA+(1-A)B +]-(AA ++(1-A)B +)_>RA +
(2 3)
is defined by (1 6b) and the #, are positive solutions of the equation (2 2)
PROOF. By (1 4) and (l 5) we have
where
[AA, + (1
A)B1] -1 > K(AA- + (1
A)Bi-1)
(24)
and
[,A + (1- A)B1] -1- (AAi-’ + (1- ,)Bi-1) >_/fAi-1
where K is defined by (16a),
PA+P"
by (16b) and the
#,
(25)
are solutions of (22)
Since
(PAP*) + (2.1) follows from (24) and (23) from (2.5)
SOME RELATED RESULTS
Let (Y, B, #) be a probability space and Au, y E Y a collection of positive semi-definite matrices of
the same order. Let Ay (a,m), 1 < i, j < n and y E Y Assume that a,ju as a function of y is
measurable for every 1 < i, j < n The following results were proved in [9,10]
Suppose there exists a set D B such that #(D) 1 and AulAyg. Au2Ayl for every yl, yg. E D
Let R(Ay) be the same for all y E D E B. Suppose Ay and
as functions of y are integrable with
3.
Au+
respect to # Then
Ay#(dy)
A+u #(dy).
<
(3 l)
By fv Au#(dy) we mean the matrix whose (i, j)th element is fv a,z#(dY).
TIIEOREM 3. If also all positive eigenvalues of Ay for all y Y are in the interval
0 < rn < M, then the following inequalities hold
+ rn)
A+u#(dy < (M4Mm
Ira, M] where
Ay#(dy)
(3 2)
and
f A+u(dY)- Il ,"
Ay#(dy)
<
Mm
I.
(33)
PROOF. As in [9], we have that there exists an orthogonal matrix C such that
where Aly, A2u,
be the rank of Av
Alu, A2u,
CTAC diag{Aw, A2y, A,w}, y E Y
A,w are the eigenvalues of Ay Since Av is positive semi-definite, each A,y >_ 0. Let k
We can assume without loss of generality that
Aku # 0
for every y E Y,
and Ak+l,y
Ak+2,y
Note that
A+
so tha
C diag
Aly A2y
Aky
.... Any
0 for every y
Y.
709
ON MA’I RIX CONVEXrrY ()F THE MOORE-PENROSE INVERSE
C
AC
diag
1
1
1
A A
A
O,
Thus, we have
where K
(M + m)9/(4Mm) The inequality
-
is the well-known Kantorovich inequality Hence each diagonal element in the above diagonal matrix is
nonnegative This completes the proof of (3 2)
Similarly,
frA#(dY) [frA,z(dy)]
where
(v/-M,
Cdiag{fr, Ai-(dU)-(frAlu#(dY))
RI
: Ak
ff,
CT
Aku#(dy)
#(dy)
-1
,I
The inequality
fyA-(yi#(dy) --/y A,y#(dy)
-1
<_
R
is a simple consequence of the following Mond-Shisha inequality 12]
f f-(f f-l) -1-< (V/-_ V/-)
where re <
f < M, 0 < re < M. Namely
1
<
1
<
1
M-f-re
so that by substituting
f ---,
,
1
we get
Thus each diagonal element in the above diagonal matrix is non-positive. This completes the proof
Moreover, we can consider the powers of A and A +. For simplicity of notation, if r < 0, we shall
use A (r) for (A+) -r. Note that (A+)
(A-r) +
and A(ur), (r < 0 < s) as
TIIEOREM 4. Let R(Au) be the same for all y E D E B. Suppose
Then
functions of y are integrable with respect to
A
A(ur)#(dy)
>_
PROOF. As in the proof of (3.2) and (3 3), we have
Au#(dy)
(3.4)
710
B MOND AND
E PE(ARI(
Each diagonal element n the above diagonal matrix is nonnegative This follows from the fact that if f
and ff are positive and integrable, the well-known inequality for means of orders s and r states that
(3 5)
which is the same as
Similar consequences of converse inequalities for (3 5) (see [12] and [13], respectively) are the next
two theorems
TIIEOREM 5. Let the conditions of Theorem 4 be satisfied and let all positive eigenvalues of Au
for all y E Y belong to the interval Ira, M] (0 < rn < M) Then the following inequality holds
A#(dy)
A")#(dy)
& A
(3 6)
where
/x
(s
r)(’),
1)
(r
s)(7
7
1)
M/m.
(3 7)
THEOREM 6. Let the conditions of Theorem 5 be satisfied Then
Au#(dy)
< AI
(3 8)
where
h
max {[OM"
0c[0,1]
+ (1 O)m"]
Of course (3 2) and (3 3) are the special cases r
[OM + (1
1, s
1 of(3
O)rn]}.
6) and (3 8)
REFERENCES
BENDAT, J and SHERMAN, S., Monotone and convex operator functions, Trans. Amer. Math.
Soc. 79 (1955), 58-71
[2] DAVIS, C, Notions generalizing convexity for functions defined on spaces of matrices, Proc.
Syrup. Pure Math., Vol 7 Convexity, Amer. Math. Soc (1963), 187-201
[3] MOORE, M H, A convex matrix function, Amer. Math. Monthly $0 (1973), 408-409
[4] OLKIN, I. and PRATT, J, A multivariate Tchebycheff inequality, Ann. Math. Statist. 29 (1958),
226-234
[5] WHITTLE, P, A multivariate generalization of Tchebychev’s inequality, Quart. J. Math. Oxford,
Ser [2] 9 (1958), 232-240
[6] HORN, R.A. and JOHNSON, C.R., Topics in Matrix Analysis, Cambridge University Press, New
York, 1991.
[7] MARSHALL, A.W. and OLKIN, I., lnequahties: Theory of Majorizatton and its Apphcattons,
Academic Press, New York, 1979.
[8] GIOVAGNOLI, A. and WYNN, H.P., G-majorization with application to matrix orderings, Lm.
Alg. Appl. 67 (1985), 111-135
[9] KAFFES, D G., An inequality for matrices, Bull. GreekMath. Soc. 22 (1981), 143-159
10] KAFFES, D G, MATHEW, T., RAO, M B and SUBRAMANYAM, K, On the matrix convexity
of the Moore-Penrose inverse and some applications, Lm. Multdin. Alg. 24 (1989), 265-271
11 MOND, B and PEtARIt, J E., Reverse forms of a convex matrix inequality, Lm. Alg. Appl., to
appear
[12] MOND, B and SHISHA, O, Difference and ratio inequalities in Hilbert space, in Inequahnes,
Vol 2, Academic Press, New York, 1970, 241-249
[13] PE(ARIt, J E and BEESACK, P R., On Knopp’s inequality for convex functions, Canad. Math.
Bull. 30 (1987), 267-272