Inequalities For Spreads Of Matrix Sums And Products

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1 Applied Mathematics E-Notes, 4(004), c ISSN Available free at mirror sites of amen/ Inequalities For Spreads Of Matrix Sums And Products Jorma K. Meriosi, Ravinder Kumar Received 5 February 004 Abstract Let A and B be complex matrices of same dimension. Given their eigenvalues and singular values, we survey develop simple inequalities for eigenvalues and singular values of A + B, AB, anda B. Here denotes the Hadamard product. As corollaries, we find inequalities for additive and multiplicative spreads of these matrices. 1 Introduction Let A be a complex n n matrix (assume n throughout) with eigenvalues λ 1,..., λ n, denoted also by λ i (A) = λ i. Order them λ 1... λ n if they are real. In the general case, order them in absolute value: λ (1)... λ (n), and denote also λ (i) (A) = λ (i).wedefine the additive spread of A by ads A =max λ i λ j i,j and multiplicative spread (assuming the λ i s nonzero) by mls A =max i,j λ i. λ j Several inequalities for the additive spread are nown (see [9] and its references). The multiplicative spread of a Hermitian positive definite matrix, the Wielandt ratio, is widely studied (see [1] and its references). Let σ 1... σ n ( 0) be the singular values of A, denoted also by σ i (A) =σ i.if A is nonsingular (i.e., if σ n > 0) we define its (spectral) condition number by cnd A =(mlsa σ A) 1 1 =. σ n It measures the numerical instability of A (see e.g. [4]). Mathematics Subject Classifications: 15A45, 15A18. Department of Mathematics, Statistics and Philosophy, FIN University of Tampere, Finland Department of Mathematics, Dayalbagh Educational Institute, Dayalbagh, Agra 8005, Uttar Pradesh, India 150

2 J. K. Meriosi and R. Kumar 151 Let 1 i<j n. If the eigenvalues of A are real, we define the additive mid-spread of A by and the multiplicative mid-spread by ads ij A = λ i λ j mls ij A = λ i (λ j > 0). λ j Assuming nothing about eigenvalues, we define the absolute multiplicative spread of A by Mls A = λ (1) (λ (n) =0) λ (n) and the absolute multiplicative mid-spread by Mls ij A = λ (i) λ (j) (λ (j) =0). We do not find the absolute additive spread interesting. Finally, we define the midcondition number of A by cnd ij A = σ i (σ j > 0) σ j (although they may be bad measures of condition). There are many well-nown inequalities for eigenvalues and singular values of A+B, AB, anda B, when the eigenvalues and singular values of A and B are given, see e.g. [], [3], [5], [6], [7], [8], [10], [11], [1]. Here denotes the Hadamard product. We will survey develop simple inequalities. We are particularly interested in their analogies. As corollaries, we will find inequalities for spreads of A + B, AB, and A B, whenthespreadsofa and B are given. There is a deep theory behind the eigenvalues of the sum of Hermitian matrices and the singular values of the product of square matrices (see e.g. [3] and its references), but our approach is elementary. Eigenvalues of A + B If A and B are Hermitian, then we can both underestimate and overestimate eigenvalues of A + B by using eigenvalues of A and B. Hence we can overestimate spreads of A + B by using spreads of A and B. THEOREM 1 (Weyl, see e.g. [], Theorem III..1; [6], Theorem 4.3.7). Let A and B be Hermitian n n matrices. If 1 i n and 1 l n i +1,then λ i+l 1 (A)+λ n l+1 (B) λ i (A + B) λ i +1 (A)+λ (B). (1) λ i (A)+λ n (B) λ i (A + B) λ i (A)+λ 1 (B) ()

3 15 Spreads of Matrices λ n (A)+λ n (B) λ n (A + B), λ 1 (A + B) λ 1 (A)+λ 1 (B). Lidsii s sum inequalities and their further developments (see e.g. [], [3]) generalize (). COROLLARY. Let A and B be Hermitian n n matrices. If 1 i<j n and 1 l n j +1,then ads ij (A + B) ads i +1,j+l 1 A +ads,n l+1 B. (3) ads ij (A + B) ads ij A +adsb ads(a + B) ads A +adsb. (4) PROOF. In ads ij (A + B) =λ i (A + B) λ j (A + B), apply the second inequality of (1) to the first term and the first inequality to the second. Then (3) follows. COROLLARY 3. Let A and B be Hermitian n n matrices and let i, j,, l satisfy the conditions of Corollary. If λ j+l 1 (A) > 0andλ n l+1 (B) > 0, then mls ij (A + B) < mls i +1,j+l 1 A +mls,n l+1 B. (5) if B is positive definite, then and if also A is positive definite, then mls ij (A + B) < mls ij A +mlsb, mls(a + B) < mls A +mlsb. (6) PROOF. Denoting α p = λ p (A), β p = λ p (B) (1 p n), r = i +1, s = j +l 1, t = n l +1,wehaveby(1) mls ij (A + B) = λ i(a + B) λ j (A + B) α r + β α s + β t. Since α r + β α r + β = α sβ + α r βt α s β t α s + β t α s β t (α s + β t ) > 0, inequality (5) follows.

4 J. K. Meriosi and R. Kumar 153 To study the additive spread of the sum of non-hermitian matrices, we recall THEOREM 4 ([9], Theorem and Lemma ). If A is a square matrix, then If A is normal, then + za ads A max adsza. (7) z =1 + za ads A =maxadsza. (8) z =1 According to Corollary, inequality (4) holds for Hermitian matrices. We extend it to normal matrices. THEOREM 5. If A and B are square matrices of same dimension, then + za ads(a + B) max adsza z =1 If A and B are normal, then +max z =1 + zb adszb. (9) ads(a + B) ads A +adsb. (10) PROOF. By (7), + B)+ z(a + B) ads(a + B) max adsz(a. z =1 Let z 0 be the maximizer. Recalling (4), we have ads z 0(A + B)+ z 0 (A + B) z0 A + z 0 A = ads ads z 0A + z 0 A + za max adsza z =1 + z 0B + z 0 B +ads z 0B + z 0 B +max z =1 + zb adszb, which proves (9). Now (8) implies (10). It is not sensible to as whether (6) can be generalized for normal matrices, since it requires that the eigenvalues are real and positive. The counterexample A = 3 0 0, B = 3 4 3, A + B = 3 4 shows that (6) cannot be generalized for matrices with real and positive eigenvalues. We have mls(a + B) = 7 but mls A +mlsb =.

5 154 Spreads of Matrices 3 Singular Values of A + B The following theorem is analogous to the second parts of (1) and (). THEOREM 6 (Fan, see e.g. [], Problem III.6.5; [7], Theorem ; [8], p. 43). Let A and B be n n matrices. If 1 i n, then σ i (A + B) σ i +1 (A)+σ (B). (11) σ i (A + B) σ i (A)+σ 1 (B) σ 1 (A + B) σ 1 (A)+σ 1 (B). The first parts of (1) and () do not have analogies for singular values. In other words: For 1 l n i +1, σ i+l 1 (A)+σ n l+1 (B) σ i (A + B) is not valid in general. A counterexample is A = I, B = I. For i = l =1,this inequality follows from the wrong inequality 9.G.1.e (also 9.G.4.b) of [8]. There does not seem to be any good way to underestimate singular values of the sum by using singular values of the summands. Certainly (11) implies σ i+l 1 (A) σ l (B) σ i (A + B), but this appears to be ineffective to our purpose. Therefore we cannot apply our methods to the additive singular value spread σ 1 (A) σ n (A). 4 Eigenvalues of AB If A and B are Hermitian and nonnegative definite, then we can both underestimate and overestimate eigenvalues of AB by using eigenvalues of A and B. Hence we can overestimate multiplicative spreads of AB by using those of A and B. (The eigenvalues of AB are real, since λ i (AB) =λ i (A 1 A 1 B)=λi (A 1 BA 1 ), and A 1 BA 1 is Hermitian. The second equality follows from the fact that if C and D are square matrices of same order, then CD and DC havethesamespectrum.) Analogously to Theorem 1, we have THEOREM 7. Let A and B be Hermitian nonnegative definite n n matrices. If 1 i n and 1 l n i +1,then λ i+l 1 (A)λ n l+1 (B) λ i (AB) λ i +1 (A)λ (B). (1)

6 J. K. Meriosi and R. Kumar 155 λ i (A)λ n (B) λ i (AB) λ i (A)λ 1 (B) (13) λ n (A)λ n (B) λ n (AB), λ 1 (AB) λ 1 (A)λ 1 (B). Lidsii s product inequalities and their further developments (see e.g. [], [3], [8], [11], [1]) generalize (13). PROOF (cf. the proof of Wang and Zhang [1], Theorem ). The second part of (1) is an easy consequence of the second part of (15). Hence, assuming B positive definite, λ i+l 1 (A) =λ i+l 1 (ABB 1 ) λ i+l 1 l+1 (AB)λ l (B 1 )=λ i (AB)λ 1 n l+1 (B), and the first part of (1) follows. If B is singular, then continuity argument applies. COROLLARY 8. Let A and B be Hermitian nonnegative definite n n matrices. If 1 i<j n, 1 l n j +1,λ j+l 1 (A) > 0, and λ n l+1 (B) > 0, then mls ij AB mls i +1,j+l 1 A mls,n l+1 B. (14) if B is positive definite, then and if also A is positive definite, then PROOF. By (1), mls ij AB mls ij A mls B, mls AB mls A mls B. mls ij AB = λ i(ab) λ j (AB) λ i +1(A)λ (B) λ j+l 1 (A)λ n l+1 (B) =mls i +1,j+l 1 A mls,n l+1 B, and (14) is proved. 5 Singular Values of AB Analogously to Theorem 7, we have THEOREM 9. Let A and B be n n matrices. If 1 i n and 1 l n i+1, then σ i+l 1 (A)σ n l+1 (B) σ i (AB) σ i +1 (A)σ (B). (15) σ i (A)σ n (B) σ i (AB) σ i (A)σ 1 (B) (16)

7 156 Spreads of Matrices σ n (A)σ n (B) σ n (AB), σ 1 (AB) σ 1 (A)σ 1 (B). (17) Gelfand s and Naimar s inequalities and their further developments (see e.g. [], [3], [8], [10], [11], [1]) generalize (16). PROOF. For the second part of (15), see e.g. [7], Theorem For the first part, proceed as in the proof of the firstpartof(1). COROLLARY 10. Let A and B be n n matrices. If 1 i < j n, 1 l n j +1,σ j+l 1 (A) > 0, and σ n l+1 (B) > 0, then if B is nonsingular, then and if also A is nonsingular, then 6 Eigenvalues of A B cnd ij AB cnd i +1,j+l 1 A cnd,n l+1 B. cnd ij AB cnd ij A cnd B, cnd AB cnd A cnd B. We have the following THEOREM 11 (see e.g. [5], Theorem 3.1; [7], Theorem 5.3.4). If A and B =(b j ) are Hermitian nonnegative definite n n matrices, then λ n (A)λ n (B) λ n (A)min b λ n (A B) and λ 1 (A B) λ 1 (A)maxb λ 1 (A)λ 1 (B). COROLLARY 1. If A and B are Hermitian positive definite matrices, then Sharper inequalities mls (A B) mls A max,l b b ll mls A mls B. λ i (A)λ n (B) λ i (A B) λ i (A)λ 1 (B), cf. (13), are not generally valid for Hermitian nonnegative definite n n matrices. For counterexample, let A = , B = = A B

8 J. K. Meriosi and R. Kumar 157 Then λ 1 (A)λ 3 (B) = 3 but λ 1 (A B) = 1, and λ (A B) =1butλ (A)λ 1 (B) =0. But can we sharpen (13) to and λ n (A)λ n (B) λ n (A)min b λ n (AB) (18) λ 1 (AB) λ 1 (A)maxb λ 1 (A)λ 1 (B) (19) for Hermitian nonnegative definite n n matrices? The first inequality of (18) and the second of (19) are elementary facts. To disprove the second inequality of (18), let A = , B = 1 Then λ (AB) =0butλ (A)min b =1. Todisprovethefirst inequality of (19), let A = , B = Then λ 1 (AB) = 7039 but λ 1 (A)max b = = Singular Values of A B For singular values, we again have some analogy. THEOREM 13 (see e.g. [5], Theorem 3.1; [7], Theorem ). Let A and B be n n matrices. If P =(p j )=(BB ) 1 and Q =(q j )=(B B) 1,then σ 1 (A B) σ 1 (A) max p max q ) 1 σ1 (A)σ 1 (B). if B =(b j ) is Hermitian nonnegative definite, then σ 1 (A B) σ 1 (A)maxb σ 1 (A)σ 1 (B). (0) The inequality σ n (A)σ n (B) σ n (A B) is not generally valid. A counterexample is A =, B = It seems that there is no good way to underestimate σ i (A B) by using singular values of A and B. Therefore we cannot find bounds for cnd ij (A B) by using condition numbers of A and B..

9 158 Spreads of Matrices 8 Eigenvalues of AB, Continued Finally, we consider absolute multiplicative spreads. It seems that the λ (i) (AB) s cannot be effectively grasped in general. Since the singular values of a normal matrix are absolute values of eigenvalues, we have by Corollary 10 the following COROLLARY 14. Let A and B be normal n n matrices such that also AB is normal. If 1 i<j n, 1 l n j +1,λ j+l 1 (A) = 0,andλ n l+1 (B) = 0, then Mls ij AB Mls i +1,j+l 1 A Mls,n l+1 B. if B is nonsingular, then and if also A is nonsingular, then Mls ij AB Mls ij A Mls B, Mls AB Mls A Mls B. If A and B commute, then analogy to Theorem 7 holds. Namely, applying Yamamoto s theorem lim σ i(a m ) 1 m = λ(i) (A) (1 i n) m (see e.g. [7], Theorem 3.3.1) and Theorem 9, we have THEOREM 15 ([7], Exercise ). Let A and B be commuting n n matrices. If 1 i n and 1 l n i +1,then λ (i+l 1) (A) λ (n l+1) (B) λ (i) (AB) λ (i +1) (A) λ () (B). (1) λ (i) (A) λ (n) (B) λ (i) (AB) λ (i) (A) λ (1) (B) λ (n) (A) λ (n) (B) λ (n) (AB), λ (1) (AB) λ (1) (A) λ (1) (B). In [7], A and B are assumed to be nonsingular in the firstpartof(1). Weproveit in the singular case. Assume that A or B is (or both are) singular. Then there exists 0 > 0 such that A = A+I and B = B+I are nonsingular for all with 0 < < 0. Because A and B commute, also A and B commute. Applying the first part of (1) to A and B, the claim follows by continuity argument. COROLLARY 16. Let A and B be commuting n n matrices. If 1 i<j n, 1 l n j +1,λ j+l 1 (A) = 0,andλ n l+1 (B) = 0,then Mls ij AB Mls i +1,j+l 1 A Mls,n l+1 B.

10 J. K. Meriosi and R. Kumar 159 if B is nonsingular, then and if also A is nonsingular, then Mls ij AB Mls ij A Mls B, Mls AB Mls A Mls B. Let A and B be normal matrices. Then AB is not necessarily normal, but it is normal if A and B commute. However, AB canbenormalevenifa and B do not commute ([6], Problem.5.9). All this motivates us to pose the following CONJECTURE. Theorem 15 remains valid if, instead of commutativity, normality of A and B (but not necessarily AB) isassumed. References [1] G. Alpargu and G. P. H. Styan, Some comments and a bibliography on the Frucht- Kantorovich and Wielandt inequalities. In Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecer, Kluwer, 000, pp [] R. Bhatia, Matrix Analysis, Springer, [3] R. Bhatia, Linear Algebra to quantum cohomology: The story of Alfred Horn s inequalities, Amer. Math. Monthly 108(001), [4] G. H. Golub and C. Van Loan, Matrix Computations, nd ed., Jonhs Hopins Univ. Pr., [5] R. A. Horn, The Hadamard Product. In Proc. Sympos. Applied Math., Vol. 40: Matrix Theory and Applications, Amer. Math. Soc., 1990, pp [6] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Pr., [7] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Pr., [8] A. W. Marshall and I. Olin, Inequalities: Theory of Majorization and Its Applications, Acad. Pr., [9] J. K. Meriosi and R. Kumar, Characterizations and lower bounds for the spread of a normal matrix, Linear Algebra Appl. 364(003), [10] R. C. Thompson, On the singular values of matrix products, Scripta Math. 9(1973), [11] B. I. Wang and B. Y. Xi, Some inequalities for singular values of matrix products, Linear Algebra Appl. 64(1997), [1] B. I. Wang and F. Z. Zhang, Some inequalities for the eigenvalues of the product of positive semidefinite Hermitian matrices. Linear Algebra Appl. 160(199),