HADAMARD PRODUCT VERSIONS OF THE CHEBYSHEV AND KANTOROVICH INEQUALITIES

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1 HADAMARD PRODUCT VERSIONS OF THE CHEBYSHEV AND KANTOROVICH INEQUALITIES JAGJIT SINGH MATHARU AND JASPAL SINGH AUJLA Department of Mathematcs Natonal Insttute of Technology Jalandhar , Punab, INDIA EMal: Receved: 10 February, 2009 Accepted: 15 Aprl, 2009 Communcated by: S. Puntanen 2000 AMS Sub. Class.: Prmary 15A48; Secondary 15A18, 15A45. Key words: Abstract: Chebyshev nequalty, Kantorovch nequalty, Hadamard product The purpose of ths note s to prove Hadamard product versons of the Chebyshev and the Kantorovch nequaltes for postve real numbers. We also prove a generalzaton of Fedler s nequalty. Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 1 of 13 Acknowledgements: The authors thank a referee for useful suggestons.

2 1 Introducton 3 2 The Chebyshev and Kantorovch Inequaltes: Matrx Versons 5 Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 2 of 13

3 1. Introducton In what follows, the captal letters A, B, C,... denote m m complex matrces, whereas the small letters a, b, c,... denote real numbers, unless mentoned otherwse. By X Y we mean that X Y s postve semdefnte X > Y mean X Y s postve defnte. For A = a and B = b, A B = a b denotes the Hadamard product of A and B. Accordng to Schur s theorem [4, Page 23] the Hadamard product s monotone n the sense that A B, C D mples A C B D. The tensor product A B s the m 2 m 2 matrx 1.1 a 11 B a 1m B... a m1 B a mm B Marcus and Khan n [10] made the smple but mportant observaton that the Hadamard product s a prncpal submatrx of the tensor product. The nequalty n 1.2 w b w a b w a holds for all a 1 a 2 a n 0, b 1 b 2 b n 0 and weghts w 0, = 1,..., n. Hardy, Lttlewood and Polya [6, page 43] attrbute ths nequalty to Chebyshev. For 0 < a a b, w 0, = 1, 2,..., n, Kantorovch s nequalty states that 2 w a + b2 1.3 w a w. a 4ab Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 3 of 13

4 In Secton 2, we state and prove matrx versons of nequaltes 1.2 and 1.3 nvolvng the Hadamard product. A generalzaton of Fedler s nequalty s also proved n ths secton. There are several generalzatons of Kantorovch and Fedler s nequalty; see [2, 3, 8, 9]. Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 4 of 13

5 2. The Chebyshev and Kantorovch Inequaltes: Matrx Versons We begn wth a Hadamard product verson of nequalty 1.2. Theorem 2.1. Let A 1 A n 0 and B 1 B n 0. Then 2.1 w A w B w A B, where w 0, = 1,..., n, are weghts. Proof. We have 2.2 w w n = w w A B w w A B = 1 2 = 1 2,=1 w A B w A w B n,=1 w w A B w w A B + w w A B w w A B n w w A A B B.,=1 Snce the Hadamard product of two postve semdefnte matrces s postve semdefnte, therefore the summand n 2.2 s postve semdefnte. Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 5 of 13

6 Our next result s a Hadamard product verson of nequalty 1.3. Theorem 2.2. Let A 1,..., A n be such that 0 < ai m A bi m, = 1,..., n here I m denotes the m m dentty matrx. Then 2.3 A for all W 0, = 1,..., n. Proof. We frst prove the nequalty A 1 a2 + b 2 2ab W W 2.4 P 1/2 AP 1/2 Q 1/2 B 1 Q 1/2 + P 1/2 A 1 P 1/2 Q 1/2 BQ 1/2 a2 + b 2 P Q, ab when 0 < ai m A, B bi m and P, Q 0. Let A = UDU and B = V ΓV wth untary U and V, and dagonal matrces D and Γ. Then A B 1 + A 1 B = U V D Γ + Γ 1 DU V a 2 + b 2 U V I m I m U V ab = a2 + b 2 I m I m, ab where the nequalty follows from 1.3. Thus we have Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 6 of P 1/2 AP 1/2 Q 1/2 B 1 Q 1/2 + P 1/2 A 1 P 1/2 Q 1/2 BQ 1/2

7 = P 1/2 Q 1/2 A B 1 + A 1 BP 1/2 Q 1/2 a2 + b 2 P Q. ab Snce the Hadamard product s a prncpal submatrx of the tensor product, the nequalty 2.4 follows from 2.5. On takng B = A and Q = P n 2.4 we see that 2.3 holds for n = 1. Further, by 2.4 we have A A 1 + A 1 for, = 1,..., n. Summng over,, we have n,=1 [ A A 1 whch mples that A A 1 A a2 + b 2 W W ab ] n a 2 + b 2 W W, ab,=1 a 2 + b 2 n n W W. 2ab The next corollary follows on takng W = w I m, = 1,..., n. Corollary 2.3. Let A 1,..., A n be such that 0 < ai m A bi m, and w 0, = 1,..., n be weghts. Then 2 a w A w A b 2 n w I m. 2ab Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 7 of 13

8 Remark 1. The case n = 1 of Corollary 2.3 s proved n [7]. The example 2 1 A =, a = 3 5, b = shows that the nequalty need not be true. A A 1 a + b2 I 2 4ab For our next result we need the followng lemma. Lemma 2.4. Let 0 r 1. Then A r + A r A + A 1 for all A > 0. Proof. Suppose that A = UΓU wth untary U and dagonal matrx Γ. Then A r + A r = UΓ r + Γ r U UΓ + Γ 1 U = A + A 1 snce x r + x r x + x 1 for any postve real number x and 0 r 1. Theorem 2.5. Let 0 α < β. Then A α for all A > 0 and W 0, = 1,..., n. A α A β A β Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 8 of 13

9 Proof. We frst prove the nequalty 2.7 A α A α + A α A α A β + for 0 α < β. Let 0 r 1. Then A r A r A = r A r = A A 1 A A 1 + A r + A r r + A A 1 A β A β A r A β A r r + A A 1 1 where the nequalty follows from Lemma 2.4. Takng r = α/β and replacng A by A β and A by A β, we have A α A β A α A β + + A α A β A α A β. Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 9 of 13 Agan usng the fact that the Hadamard product s a prncpal submatrx of the tensor

10 product, the precedng nequalty mples 2.7. Summng over, n 2.7, we have A α A α A β A β. A α Corollary 2.6. Let 0 α < β. Then for all A > 0, = 1,..., n. =1 A α A β Proof. Takng W = I m n Theorem 2.5 we get the desred result. =1 A β Corollary 2.7. Let 0 β. Then I m A β A β for all A > 0 and W 0, = 1,..., n, where n W = I m. Proof. Takng α = 0 n Theorem 2.5 gves the desred nequalty. Remark 2. Corollary 2.7 s another generalzaton of Fedler s nequalty [5] Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 10 of 13 A A 1 I m.

11 Next we prove a convexty theorem nvolvng the Hadamard product. Theorem 2.8. The functon ft = A 1+t B 1 t + A 1 t B 1+t s convex on the nterval [ 1, 1] and attans ts mnmum at t = 0 for all A, B > 0. Proof. Snce f s contnuous we need to prove only that f s md-pont convex. Note that for A, B > 0 and s, t n [ 1, 1] the matrces A 1+s+t A 1+s A 1 s+t A A 1+s A 1+s t, 1 s A 1 s A 1 s t, B 1+s+t B 1+s B 1 s+t B B 1+s B 1+s t, 1 s B 1 s B 1 s t are postve semdefnte. Hence the matrx A X= 1+s+t B 1 s+t + A 1 s+t B 1+s+t A 1+s B 1 s + A 1 s B 1+s A 1+s B 1 s + A 1 s B 1+s A 1+s t B 1 s t + A 1 s t B 1+s t s postve semdefnte. Smlarly, the matrx A Y = 1+s t B 1 s t + A 1 s t B 1+s t A 1+s B 1 s + A 1 s B 1+s A 1+s B 1 s + A 1 s B 1+s A 1+s+t B 1 s+t + A 1 s+t B 1+s+t s postve semdefnte. Hence fs + t + fs t 2fs 2.8 X + Y = 2fs fs + t + fs t s postve semdefnte, whch mples that Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 11 of 13 fs 1 [fs + t + fs t]. 2

12 Ths proves the convexty of f. Further, note that ft = f t. Ths together wth the convexty of f mples that f attans ts mnmum at 0. Corollary 2.9. The functon gt = A t B 1 t + A 1 t B t s decreasng on [0, 1/2], ncreasng on [1/2, 1], and attans ts mnmum at t = 1 2 for all A, B > 0. Proof. The proof follows on replacng A, B by A 1/2, B 1/2 and t by 1+t 2 n Theorem 2.8. A norm on m m complex matrces s called untarly nvarant f UXV = X for all untary matrces U, V. If A s postve semdefnte and X s any matrx, then A X max a X for all untarly nvarant norms [1]. Thus the proof of the followng corollary follows from Corollary 2.9 usng the fact that g1/2 gt g1 = g0. Corollary Let 0 t 1. Then, 2 A 1/2 B 1/2 A t B 1 t + A 1 t B t A + B for all untarly nvarant norms and all A, B > 0. Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 12 of 13

13 References [1] T. ANDO, R.A. HORN AND C.R. JOHNSON, The sngular values of the Hadamard product: A basc nequalty, Lnear Multlnear Algebra, , [2] J.K. BAKSALARY AND S. PUNTANEN, Generalzed matrx versons of the Cauchy-Schwarz and Kantorovch nequaltes, Aequatones Math., , [3] R.B. BAPAT AND M.K. KWONG, A generalsaton of A A 1 I, Lnear Algebra Appl., , [4] R. BHATIA, Matrx Analyss, Sprnger Verlag, New York, [5] M. FIEDLER, Über ene Unglechung für postv defnte Matrzen, Math. Nachrchten, , [6] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequaltes, Cambrdge Unversty Press, Cambrdge, [7] J. MIĆIĆ, J. PECARIC AND Y. SEO, Complementary nequaltes to nequaltes of Jensen and Ando based on the Mond-Pečarć method, Lnear Algebra Appl., , [8] A.W. MARSHALL AND I. OLKIN, Matrx versons of the Cauchy and Kantorovch nequaltes, Aequatones Math., , [9] M. SINGH, J.S. AUJLA AND H.L. VASUDEVA, Inequaltes for Hadamard product and untarly nvarant norms of matrces, Lnear Multlnear Algebra, , [10] M. MARCUS AND N.A. KHAN, A note on Hadamard product, Canad. Math. Bull., , Hadamard Product Versons Jagt Sngh Matharu and Jaspal Sngh Aula vol. 10, ss. 2, art. 51, 2009 Ttle Page Page 13 of 13