Yuki Seo. Received May 23, 2010; revised August 15, 2010

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1 Scietiae Mathematicae Japoicae Olie, e-00, A GENERALIZED PÓLYA-SZEGÖ INEQUALITY FOR THE HADAMARD PRODUCT Yuki Seo Received May 3, 00; revised August 5, 00 Abstract. I this paper, we show a geeralized Pólya-Szegö iequality for the Hadamard product: Let A ad B be k k-positive defiite matrices such that mi A, B MI for some scalars 0 <m<m. The p (A x, x)(b x, x) k M + m for every vector x, where I is the idetity matrix ad the symbol is the Hadamard product. Itroductio. Let M k = M k (C) deote the space of k k complex matrices. For a pair A, B of Hermitia matrices the order relatio A B meas as usual that A B is positive semidefiite. I particular, A>0 meas that A is positive defiite. For A =(a ij ) ad B =(b ij ), their Hadamard product is the k k matrix of etrywise products A B =(a ij b ij ). It is commutative ulike the usual matrix product: A B = B A. The diagoal matrix formed a matrix A ca be obtaied by Hadamard multiplicatio with the idetity matrix A I. As Stya poited out i [7], the most widely used ad possibly most importat result cocerig the Hadamard product is as follows: Theorem A (Shur). If A i is positive defite (i =,,,), the so is A A A. It is likely that may matrix iequalities for the Hadamard product is based o this fact. For example, Ado [] showed the followig Cauchy-Schwarz iequality for the Hadamard product: If A i is positive defiite (i =,,,, ), the (.) A A A (A I) (A I) (A I). I fact, i the case of =,ifa ad B are diagoal matrices, the we have the Cauchy- Schwarz iequality: a i b i (.) b i. 000 Mathematics Subject Classificatio. 5A45, 5B48 ad 47A63. Key words ad phrases. Positive defiite matrix, Hadamard product, Specht ratio, Pólya-Szegö iequality, Cauchy-Schwarz iequality. a i

2 4 Y. SEO I [4], Pólya-Szegö showed a reverse of Cauchy-Schwarz iequality (.): If the real umbers a i ad b i (i =,,,) satisfies the coditio (.3) 0 <m a i,b i M for i =,,, the (.4) a i b i M + m a i b i. I [], Grueb-Rheiboldt poited out that the Pólya-Szegö iequality is a direct specializatio of the followig iequality which is equivalet to the Katorovich iequality: If {a i } ad {b i } (i =,, ) are two sequeces of real umbers with the coditio (.3) ad {ξ i } deotes aother sequece with ξ i <, the a i ξ i b i ξ i M + m (.5) a i b i ξi. From this viewpoit, Grueb-Rheiboldt showed a geeralized form of the iequality (.5), which is called a geeralized Pólya-Szegö iequality: Let A ad B be commutig positive defiite matrices such that mi A, B MI for some scalars 0 <m<m. The (A x, x)(b x, x) M + m (.6) (ABx, x) I this paper, we show a geeralized Pólya-Szegö iequality for the Hadamard product ad a reverse iequality of -variables of the Cauchy-Schwarz oe (.) due to Ado. Hadamard product versio The tesor product M k M k of copies of M k is idetified with M k i a atural way. It has bee kow that the Hadamard product is a pricipal square submarix of the tesor product. This fact is formulated as follows: Lemma. For each positive iteger there is a ormalized positive liear map Φ from the ope coe of positive defiite matirces i M k to oes i M k that satisfies Φ (A A )=A A for all A i M k ad i =,,. To prove our mai results, we eed the followig well-kow two lemmas. We give a proof for coveiece. Lemma ([3, 6]). If A is a positive defiite matrix i M k, the A I k A. Proof. Let P be the matrix with all etries. We defie a liear map by Φ X (A) =A X for a fixed X. The it follows that Φ X is positive if ad oly if X is positive semidefiite. If we put X = I λp, the we have the desired iequality sice k = max{λ : I λp 0}. Lemma 3. Let A ad B be commutig positive defiite matrices such that mi A, B MI for some scalars 0 <m<m. The A + B M + m AB.

3 A GENERALIZED PÓLYA-SZEGÖ INEQUALITY 43 Proof. Put C = A B ad it follows that C is positive defiite ad m M I C M m I. The ( M m I C)(C m M I) 0 implies I + C M + m C ad hece we have A +B M +m AB. We show a geeralized Pólya-Szegö iequality for the Hadamard product. Theorem 4. Let A ad B be k k-positive defiite matrices i M k such that mi A, B MI for some scalars 0 <m<m. The (A x, x)(b x, x) k M + m Proof. Sice A I ad I B are commutative ad mi I A I,I B MI I, it follows from Lemma 3 that (A I) +(I B) M + m (A I)(I B) + m =M (A B). From Lemma we have A I + I B M + m A B. Therefore, by the arithmetic-geometric mea iequality ad Lemma we have (A x, x)(b x, x) ((A x, x)+(b x, x)) k (((A I)x, x)+((b I)x, x)) ( = k ( A I + B ) I )x, x k M + m Remark 5. The iequality (A x, x)(b x, x) M +m does ot hold i geeral. I fact, put ( ) ( ) ( 3 0 A =, B = ad x = 0 ) ad I A, B 3I. The we have (A x, x)(b x, x) =6 5= O the other had, we have M +m = 5 3 8=3.33. Therefore, (A x, x)(b x, x) M + m.

4 44 Y. SEO 3 -variables versio We recall the Specht ratio: As a reverse of the arithmeticgeometric mea iequality, Specht [5] estimated the ratio of the arithmetic mea to the geometric oe: For x,,x [m, M] with 0 <m<m, x + + x (3.) S(h) x x where h = M m ad S(h) is defied for h as (h )h h (3.) S(h) = (h >) ad S() =. e log h The followig lemma is regarded as a reverse of the arithmetic-geometric mea iequality for the Hadamard product: Lemma 6. Let A i be positive defiite matrices i M k such that mi A i MI for some scalars 0 <m<m ad i =,,,,. Put h = M m. The (A I + + A I) S(h)(A A ). Proof. Sice A I I,,I I A are mutually commutative ad the spectrum is cotaied i [m, M], by the Specht theorem (3.) it follows that (A I I + + I I A ) S(h) (A I I) (I I A ) = S(h)(A A ) = S(h)(A A ) ad hece from Lemma (A I I + I I A ) S(h)(A A ). Therefore, we have (A I + + A I) S(h)(A A ). Now, we show -variables versio of Theorem 4: Theorem 7. Let A i be positive defiite matrices i M k such that mi A i MI for some scalars 0 <m<m ad for i =,,,,. Put h = M m. The (A x, x)(a x, x) (A x, x) k S(h )(A A A x, x) for every vector x, where the Specht ratio S(h) is defied by (3.). Proof. By Lemma ad Lemma 6, it follows that (A x, x) (A x, x) ((A x, x)+ +(A x, x)) k ((A I)x, x)+ +((A I)x, x) ( ) = k (A I + + A I)x, x k S(h )(A A x, x)

5 A GENERALIZED PÓLYA-SZEGÖ INEQUALITY 45 Remark 8. I the case of =, Theorem 4 is more precise estimates tha Theorem 7. I fact, Yamazaki [8] poited out that M + m = h + S(h ) for h = M h m. Fially, we show a -variables Pólya-Szegö type iequality for the Cauchy-Schwarz oe (.) for the Hadamard product due to Ado: Theorem 9. Let A i be positive defiite matrices i M k such that mi A i MI for some scalars 0 <m<m ad for i =,,,,. Put h = M m. The (A I) (A I) S(h )(A A ), where the Specht ratio S(h) is defied by (3.). Proof. By the arithmetic-geometric mea iequality ad Lemma 6, it follows that (A I) (A I) (A I + + A I) S(h)(A A ). Replacig A i by A i, we have the desired iequality. Refereces [] T.Ado, Cocavity of certai maps o positive defiite matrices ad applicatios to Hadamard products, Liear Algebra Appl., 6 (979), [] W. Greub ad W. Rheiboldt, O a geeralizatio of a iequality of L.V.Katorovich, Proc. Amer. Math. Soc., 0 (959), [3] V.I.Paulse, S.C.Powe ad R.R.Smith, Schur products ad matrix completios, J.Fuctioal Aal., 85 (989), [4] G.Pólya ad G.Szegö, Aufgabe ud Lehrsätze der Aalysis,, Berli, 95. [5] W. Specht, Zur Theorie der elemetare Mittel, Math. Z., 74 (960), [6] D. Stojaoff, Idex of Hadamard multiplicatio by positive matrices, Liear Algebra Appl., 90 (999), [7] G.P.H.Stya, Hadamard Products ad Multivariate Statistical Aalysis, Liear Algebra Appl., 6 (973), [8] T. Yamazaki, A extesio of Katorovich iequality to -operators via the geometric mea by Ado-Li-Mathias, Liear Alg. Appl., 46 (006) Faculty of Egieerig, Shibaura Istitute of Techology, 307 Fukasaku, Miumaku, Saitama-City, Saitama , Japa. address : yukis@sic.shibaura-it.ac.jp