SIAM J. MATRIX ANAL. APPL.Vol. 14, No. 1, pp. 132-136, January 1993

(C) 1993 Society for Industrial and Applied Mathematics012

MORE MATRIX FORMS OF THE ARITHMETIC-GEOMETRICMEAN INEQUALITY*

RAJENDRA BHATIA AND CHANDLER DAVIS:

Dedicated to T. Ando on his 60th birthday

Abstract. For arbitrary n n matrices A, B, X, and for every unitarily invariant norm, it is proved that2111A XB --< AA X / XnB* III.

Key words, geometric mean, singular values, unitarily invariant norm

AMS(MOS) subject classifications. 15A42, 15A60, 47A30, 47B05, 47B10

In an earlier paper [3] it was proved that, for arbitrary n n matrices A and B,

(1) 2sj( A * B < sj( AA * + BB * ), j 1, 2,..., n,

where s9 are the singular values in decreasing order. This means, in particular, that

(2) 2 III A * B !11 --< Ill AA * + BB * IIIfor every unitarily invariant norm.

Our main result is a considerable strengthening of the latter inequality.THEOREM 1. For arbitrary n n matrices A, B, X,

(3) 2IIIA*XB]II <= IIIAA*X / gOB*Ill

for every unitarily invariant norm.(The corresponding strengthening of statement is easily seen not to hold, even

for positive-definite 2 2 matrices.)An incidental benefit from our proof is the insight it may afford into the order of

the factors in and (2). That the order is critical was shown in [3]; Theorem maymake it look less surprising.

For the special case of the bound norm, (3) was discovered earlier by McIntosh 6with a different proof (and a different motivation). Fuad Kittaneh found the Hilbert-Schmidt case of (3), with conditions for equality, before we began our work. He alsomade subsequent contributions, which are being published elsewhere.

COROLLARY. Forpositive semidefinite matrices A, C, andfor any unitarily invarianttlorm,

IIIAC2A III --< IliA 2c2 I1[.

(The case of the p-norm, with p a power of 2, is a known inequality 7 ].) This followsfrom Theorem by

I[IAC2AI[I-< 1/211IA2C2 / CZAZlll __< IIIA2C2111.The theorem can be strengthened in case A and B are positive semidefinite.

Received by the editors April 11, 1990; accepted for publication (in revised form) May 9, 1991.t Indian Statistical Institute, 7, SJS Sansanwal Marg, New Delhi 110 016, India (vikram!

isid!rbh@ncst.ernet.in).{Department of Mathematics, University of Toronto, Toronto M5S 1A1, Canada (davis@

math.toronto.edu ).

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ARITHMETIC-GEOMETRIC MEAN INEQUALITY 133

THEOREM 2. For n n matrices A, B, X with A >= 0, B > 0, andfor any unitarilyinvariant norm, the realfunction(4) f(p) ][IA +PXB ’-p + A ’-PXBis convex on [-1, 1] and takes its minimum at p O.

Another result, akin to Theorem but with a separate proof, is proved in Theo-rem 3.

THEOREM 3. Fix an n n matrix G, and any unitarily invariant norm. Amongchoices ofF and H which make the matrix

positive semidefinite, that which minimizes its norm is F G*I, H GI.customary, GI denotes (G’G)1/.)

This is an especially simple example of efficient completion of a partial matrix [4],5 ], in this case of

G* ?

It differs from many other situations in that here the same completion is efficient formany norms; contrast [4 ].

Proofof Theorem 1. The main features already appear in a particular case.Case 1. B A > O, X >= 0. First take the bound norm II. It is to be proved that

2[IAXAll <= IlAuX / XA21l.With both sides depending continuously upon A, it is enough to treat the case where Ais invertible. Then, introducing the notation T AZx A(AXA)A- we note that(T) ( T* (AXA); the norm of 2AXA is to be compared with that of T + T*.

Now, AXA is positive, so its norm is the maximum among its eigenvalues, say X.In light of 3 (T)

__W(T), the numerical range of T, there is some unit vector x

with x*Tx X (it could be an eigenvector belonging to eigenvalue X1, but it need notbe). For the same x, x* T*x k kl, SO that

211AXA 2) x * T + T* x <= T + T*

as desired.We go from this to the assertion for arbitrary unitarily invariant norms by a familiar

procedure. First, it is enough to obtain the conclusion for the kth Ky Fan norm Ilk forarbitrary k 1, 2 n; see [2, 7]. We have now done it for

By AkA denote the operator A (R) (R) A (k factors) restricted to its invariantsubspace 3.*(Cn) consisting of all skew k-tensors; by Ttkl denote the operator T (R) (R)

(R) + (R) T(R) (R) + + (R) (R) (R) T restricted to itsinvariant subspaceA*(Cn), and similarly (AXA)II; see [2, 6]. Now TIk (3*A)(AXA)L(A*A)-I, sor( TIk) cr((AXA)tkl) and the reasoning used above for AXA and T can be invoked.Remember that the eigenvalues of (AXA) are obtained from the eigenvalues X >=)k2 ofAXA as all sums XI + + Xik formed using distinct indices i,..., i,, andremember that (AXA) O. So the conclusion is that for some unit vector y of 3* C ),

2]IAXAI[ 2(3 + +()

21](AXA)I[I y*(T + (Tt*)*)y <= ]I(T + T*)t*[I.

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134 RAJENDRA BHATIA AND CHANDLER DAVIS

On the right we have the norm of a Hermitian matrix, in particular, the modulus of oneof its eigenvalues. But we know these are sums of eigenvalues of T + T*; hence, theright-hand member of (5) is less than or equal to the sum of the top k singular values ofT + T*, viz., T + T* Ilk. Case is complete.

We turn to the general case. This is proved by reducing it, not to Case 1, but to aform where the same ideas can be tricked into serving.

First, consider the polar resolutions A A1V, B BI W (with A and B positivesemidefinite and V and W unitary). On the right-hand side in (3), we are consideringAA*X + XBB* A2X + XB2, while on the left we are considering A*XBV*AXB W, which has the same norms as AXBI. Therefore, the general case will beestablished if we prove the result for A >= 0, B >- 0, X arbitrary. As before, we are alsofree to assume that A and B are invertible.

We begin by defining the operators

A 0A=

0 B>_-0, X=

* 0

With this definition,

{6)

0 AXB ]AoXoAoBX *A 0

0A }Xo + XoA }

BZx . + X .A 2

A2X -1- XB2 ]0

These matrices have eigenvalues that come in pairs Thus we may write the eigenvaluesof AoXoAo as X >= X2 kn --n --kl, where the hj are the singularvalues of AXB, and the eigenvalues of AoX} + XoA} as +gj., where g > >= tsn arethe singular values ofA 2X + XB2.

To adapt the ideas of Case 1, we will estimate the quantity (AoXoAo) Ik] (1 =<k =< n). As the norm of a Hermitian operator, it is given by an eigenvalue, hence by asum ofsome +X. Choosing these to maximize the modulus ofthe sum requires choosingall to have the same sign; that is, [](AoXoAo)I]]l + + X [[AXBI]. Similarlyfor the other matrix in (6).

Now let T A}Xo Ao(AoXoAo)A Its spectrum is the same as that ofAoXoAo.We must consider the second matrix in (6), which is T + T*. Reasoning as in Case 1,there is some unit vector z A(C2n) such that

2IIAXBII- 2(h + + hk)= z*(T + T*)I]z

<-II(T+ T*)[]II + + = [IAzX+XB2}I.

That is, 2II[AXBI[] <- [[IA2X-t- XB21]I holds for all the Ky Fan norms, whence it holds forall unitarily invariant norms. U]

We have an alternative proof of Theorem which is in some respects preferable.In particular, the recourse to tensor algebra can be avoided. We sketch the key part, theproof that for A A * and X X *, we have that

k k

(7) 2 s(AXA) <- s(AzX+ XA2).j=l j=l

Assume without loss of generality that A is invertible. Then, letting T A2X sothat T* XA 2, we have that T A(AXA)A- and T* A- (AXA)A whence the three

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ARITHMETIC-GEOMETRIC MEAN INEQUALITY 135

matrices AXA, T, and T* all have the same spectrum. Denote these eigenvalues by X l,

X2, in order of decreasing Xjl. Because AXA is Hermitian, its singular values aresj ]Xj], so that the left-hand member of (7) is just

k

j=l

When T is put in Schur upper-triangular form with diagonal (Xl, k2, ...), we arereferring it to an orthonormal basis, (el, e2, en), for which

(T* takes lower-triangular form relative to this basis.) Thereforek k k

2 Ixjl ]e2(r+ T*)ejl N sj(T+ T*),j=l j=l j=l

the inequality by Ky Fan’s variational criterion [2 ]. This establishes (7).ProofofTheorem 2. Again, by continuity, we may assume that A and B are invertible.

Since f is plainly continuous andf(-p) f(p), both conclusions of the theorem willfollow if we now show that 2f(p) _-< f(p + q) + f(p q) when p + q both lie in[-1, II.

Consider the mapping on positive matrices defined by

2///p(Y) APYB-p + A-PYBp.

Theorem tells us that III.//gu(Y)[ll >_- III Y[II. Apply this to Y /tp(AXB), using theidentity 2’/[q(/I/[p(Y)) /[/[p+q(Y) nt- ’gp_q(Y), and the result is

2f(p) 4 III/tp(AXB)III ----< 4 [II//q(J/Ip(AXB))III

<= 21]I//tp+u(AXB)II[ + 2111J/tp_u(AXB)II] f(p + q) + f(p- q). [--1

Proofof Theorem 3. Consider the polar resolution G UK with U unitary. Thenthe matrix under study,

G*

has the same norms as its unitary transform

0 G* H 0 K

This, in turn, has the same norms as its unitary transform

[0 1]0 K 0 K

Without increasing any norm, we can go from one of these to their mean. [] whereM 1/2 (H + L). This positive-semidefinite matrix has the same norm as its unitarytransform (by the unitary 2-1/2[I 21])

0 M-K

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136 RAJENDRA BHATIA AND CHANDLER DAVIS

This evidently satisfies

M+Ko M-KO]>--[2K 0]>0’00

which is known to imply

M+K 0 ]0 M-K

But this comparison matrix [2ff ] is just what matrix

M+K 0 ]0 M-K

reduces to when, in particular, F [G*[, H [G[, H L K M. Therefore, undoingall the unitary similarities leads to the inequality announced.

The reason we see this result as related to Theorem may be seen in the specialcase in which G >= 0 is invertible. Those pairs (F, H) that retain positivity of[ ]/] whileminimizing the rank are those obtained by H GF-G. Such pairs all have the propertythat G F # H, the geometric mean of F and H ]. The most efficient of them, insome sense, should be (G, G). Theorem 3 gives one sense in which this holds. Anotheris that F + GF-G >_- 2G, a familiar elementary computation. (Taking traces in thisinequality gives Theorem 3 in the special case of the trace norm.)

Acknowledgments. Davis thanks the Indian Statistical Institute and the NaturalScience and Engineering Research Council (Canada) for making possible a visit duringwhich this work was done. We thank two referees for correcting our faulty version ofTheorem 2 and for contributing the corollary.

REFERENCES

T. ANDO, Concavity ofcertain maps on positive definite matrices and applications to Hadamard products,Linear Algebra Appl., 26 (1979), pp. 203-241.

[2] R. BHATIA, Perturbation Bounds for Matrix Eigenvalues, Longman, Essex and Wiley, New York, 1987.[3] R. BHATIA AND F. KITTANEH, On the singular values of a product of operators, SIAM J. Matrix Anal.

Appl., 11 (1990), pp. 272-277.[4] C. DAvIs, An extremal problem for extensions of a sesquilinear form, Linear Algebra Appl., 13 (1976),

pp. 91-102.5 C. DAVIS, W. M. KAHAN, AND H. F. WEINBERGER, Norm-preserving dilations and their applications to

optimal error bounds, SIAM J. Numer. Anal., 19 (1982), pp. 445-469.[6 A. McINTOSH, tteinz inequalities and perturbation ofspectralfamilies, Macquarie Mathematics Reports

79-0006, 1979.[7] C. J. THOMPSON, Inequality with applications in statistical mechanics, J. Math. Phys., 6 (1965), pp. 1812,

1813.

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