Norm Inequalities of Positive Semi-Definite Matrices

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1 Norm Inequalities of Positive Semi-Definite Matrices Antoine Mhanna To cite this version: Antoine Mhanna Norm Inequalities of Positive Semi-Definite Matrices 15 <hal-11844v1> HAL Id: hal Submitted on 14 Aug 15 (v1, last revised 14 Sep 15 (v5 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not The documents may come from teaching and research institutions in France or abroad, or from public or private research centers L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés

2 Norm Inequalities of Positive Semi-Definite Matrices Antoine Mhanna 1 1 Dept of Mathematics, Lebanese University, Hadath, eirut, Lebanon tmhanat@yahoocom Abstract For positive block-matrix we write M = X M + n+m, with A M+ n, M + m The main result is first to study the consequences of a decomposition lemma due to C ourrin and second to extend the class of these PSD matrices M written by blocks that satisfies the inequality: M A+ for all symmetric norms and to give examples whenever it is necessary Keywords : Matrix Analysis, Hermitian matrices, symmetric norms 1 Introduction LetAbeann nmatrixandf anm mmatrix, (m > nwrittenbyblockssuchthatais adiagonalblockandallentriesotherthanthoseofaarezeros, thenthetwomatriceshave the same singular values and for all unitarily invariant norms A = F = A, we say thenthat thesymmetricnormon M m inducesasymmetricnormonm n, soforsquare matrices wemay assumethat ournormsaredefinedonall spaces M n,n 1 Thespectral norm is denoted by s, the Frobenius norm by (, and the Ky Fan k norms by k Let M + n denote the set of positive and semi-definite part of the space of n n complex matrices and M be any positive block-matrices; that is, M = X M + n+m, with A M + n, M+ m 1

3 Decomposition of block-matrices Lemma 1 For every matrix M in M + n+m written in blocks, we have the decomposition: ( ( ( A X A X = U U +V V for some unitaries U,V M n+m Proof Factorize the positive matrix as a square of positive matrices: ( ( ( A X C Y C Y X = Y D Y D we verify that the right hand side can be written as T T +S S so : ( ( ( ( ( ( C Y C Y C C Y Y Y D Y = D Y + D Y D }{{}}{{}}{{}}{{} T T S S Since TT = ( CC +YY = ( A, SS = ( Y Y +DD AA is unitarily congruent to A A for any square matrix A, the lemma follows = ( and Remark 1 As a consequence of this lemma we have: M A + for all symmetric norms Equations involving unitary matrices are called unitary orbits representations Recall that if A M n, R(A = A+A and I(A = A A i Corollary 1 For every matrix in M + n written in blocks of the same size, we have the decomposition: ( ( A X A+ X = U R(X U +V ( A+ V +R(X for some unitaries U,V M n Proof Let J = 1 ( I I where I is the identity of M I I n, J is a unitary matrix, and we have: ( A+ A A X J X J R(X + X X = A X X A+ +R(X }{{} N

4 Now we factorize N as a square of positive matrices: ( ( ( A X X = J L M L M M F M J F and let: A direct computation shows that: δψ = 1 ( ( L M δ = J M = L+M 1 M +F F ( ( M L F M L M ψ = M J = L+M M L 1 F F +M F M (L+M (L+M+(M+F(F+M (L+M (M L+(M+F(F M (M L(L+M+(F M(F+M (M L(M L+(F M(F M = Γ Γ+Φ Φ (1 where: Γ = 1 ( L+M M L, and Φ = 1 ( F +M F M to finish notice that for any square matrix A, A A is unitarily congruent to AA and, ΓΓ, ΦΦ have the required form The previous corollary implies that A+ R(X and A+ R(X Corollary For every matrix in M + n written in blocks of the same size, we have the decomposition: X = U ( A+ +I(X U +V ( A+ V I(X for some unitaries U,V M n+m ( ( A X A Proof The proof is similar to Corollary 1, we have: J 1 X J 1 = ix ( I where J 1 =, and ii K = JJ 1 X J 1J = A+ +I(X A+ I(X ix here (* means an unspecified entry, the proof is similar to that in Corollary 1 but for reader s convenience we give the main headlines: first factorize K as a square of positive matrices; that is, M = X = J 1 J L JJ 1 next decompose L as in Lemma 1 to obtain M = J 1J (T T +S SJJ 1 = J 1J (T TJJ 1 +J 1J (S SJJ 1 3

5 ( A+ where TT = +I(X property completes the proof and SS = ( A+ finally the congruence I(X The existence of unitaries U and V in the decomposition process need not to be unique as one can take the special case; that is, M any diagonal matrix with diagonal ( ( k k entries equals a nonnegative number k, explicitly M = ki = U I U +V I V for any U and V unitaries Remark Notice that from the Courant-Fischer theorem if A, M + n, then the eigenvalues of each matrix are the same as the singular values and A = A k k, for all k = 1,,n, also A < = A k < k, for all k = 1,,n Corollary 3 For every matrix in M + n written in blocks of the same size, we have: X 1 { U for some unitaries U,V M n+m ( A+ + X X ( } U +V V A+ + X X Proof This a consequence of the fact that I(X I(X 3 Symmetric Norms and Inequalities In [1] they found that if X is hermitian then M A+ ( for all symmetric norms It has been given counter-examples showing that this does not necessarily holds if X is a normal but not Hermitian matrix, the main idea of this section is to give examples and counter-examples in a general way and to extend the previous inequality to a larger class of PSD matrices written by blocks satisfying ( Theorem 31 If A and are positive definite matrices of same size Then X > A X 1 X 4

6 ( ( ( A X I X 1 A X Proof Write X = 1 X ( I I X 1 I the identity matrix, and that complete the proof since for any matrix A, where I is A X AX, X ( A Theorem 3 Let M = be any square matrix written by blocks of same size, C D if AC = CA then det(m = det(ad C Proof Suppose first that A is invertible, let us write M as M = ( ( Z I E V I F (3 upon calculation we find that: Z = A, V = C, E = A 1, F = D CA 1 taking the determinant on each side of (3 we get: det(m = det(a(d CA 1 = det(ad C the result follows by a continuity argument since the Determinant function is a continuous function Given thematrix M = X amatrix inm + n written byblocks of samesize, we know that it M is not PSD, to see this notice that all the extracted principle submatrices of M are PSD if and only if X = and A is positive semi-definite Even if a proof of this exists and would take two lines, it is quite nice to see a different constructive proof, a direct consequence of Lemma 1 Theorem 33 Given X a matrix in M + n written in blocks of same size: 1 If X is positive semi-definite, I(X > or I(X <, then there exist a ( A Y matrix Y such that M = Y is positive semi-definite and: ( A Y Y > A (4 for all symmetric norms 5

7 ( X If X is positive semi-definite, I(X > or I(X < then there exist a ( Y matrix Y such that M = Y is positive semi-definite and: The same result holds if we replaced I(X by R(X because congruent to X ( Y Y > (5 ( A ix ix is unitarily Proof Without loss of generality we can consider I(X > cause X and ( A X X are unitarily congruent, we will show the first statement as the second one has a similar proof, from Corollary we have: X U ( A U +U ( I(X U +V ( A V ( ( A X A lx Since X is congruent to L = lx for any l C, L is PSD A is a fixed ( A ( matrix, we have U k U +V A V = β A k for some β 1 finally we set Y = lx where l R is large enough to have M k > A k, k thus M > A for all symmetric norms Notice that there exist a permutation matrix P such that P X P 1 = ( X andsincei(x > ifandonlyifi(x X A <,thetwoassertions oftheorem33 are equivalent up to a permutation similarity Corollary 31 If M = X, A a positive semi-definite matrix, and we have one of the following conditions: 1 R(X > R(X < 3 I(X > 6

8 4 I(X < Then M can t be positive semi-definite Proof y Remark 1 any positive matrix M written in blocks must satisfy M A + for all symmetric norms which is not the case of the matrix M constructed in Theorem 33 Finally we get: Theorem 34 If X and =, A, the matrix M = positive semi-definite X cannot be Proof Suppose the converse, so M = X is positive semi-definite, without loss of generality the only case we need to discuss is when R(X has positive and negative eigenvalues, by Corollary 1 we can write: M = U for some unitaries U,V M n ( A R(X ( U +V A +R(X V Now if R(X has α the smallest negative eigenvalue R(X + (α + ǫi > consequently the matrix ( A H = U R(X ( U +V A V (6 +(α+ǫi +R(X+(α+ǫI ( A+(α+ǫI X +(α+ǫi = (X +(α+ǫi (7 is positive semi-definite with R(Y >, where Y = X +(α+ǫi, by Corollary 31 this is a contradiction A natural question would be how many are the nontrivial PSDmatrices written by blocks? The following lemma will show us how to construct some of them Lemma 31 Let A and be any n n positive definite matrices, then there exist an ( ta X integer t 1 such that the matrix F t = X is positive definite t 7

9 Proof Recall from Theorem 31 that F 1 is positive definite if and only if A > X 1 X, which is equivalent to x Ax > x X 1 X x for all x C n Set f(x := x Ax and g(x := x X 1 X x and let us suppose, to the contrary, that there exist a vector z such that f(z g(z since f(x and g(x are homogeneous functions of degre d = over R if f(x g(x for all x such that x s = 1 then f(x g(x for any x C n So let us set K = max g(x, and L = min f(x since g(x and f(x are continuous functions x s =1 x s =1 and {x; x s = 1} is compact, there exist a vector w respectively v such that K = g(w, respectively L = f(v Now choose t 1 such that tf(v > g(w, to obtain t x (tax v (tav > w X(t 1 X w x X(t 1 X x for all x such that x s = 1, thus x (tax > x X(t 1 X x for any x C n which completes the proof Theorem 35 Let A = diag(λ 1,,λ n, = diag(ν 1,,ν n and M = X a given positive semi-definite matrix If X commute with A and X X equals a diagonal matrix, then M A+ for all symmetric norms The same inequality holds if X commute with and XX is diagonal Proof From Corollary?? it suffices to prove the inequality for the Ky Fan k norms ( In k = 1,,n Let P = where I I n n is the identity matrix of order n, since ( ( ( X A X A X = P X A X P 1 and X have same singular values, we will discuss only the first case; that is, when X commute with A and X X is diagonal, as the second case will follows Let D := X X = Theorem 3 we conclude that the eigenvalues of d 1 d ( A X X d n det((a µi n ( µi n D =, as X commute with A, from are the roots of 8

10 Equivalently the eigenvalues are all the solutions of the n equations: (λ 1 µ(ν 1 µ d 1 = (λ µ(ν µ d = (λ 3 µ(ν 3 µ d 3 = (λ n µ(ν n µ d n = Each equation is of nd degree, if we denote by a i and b i the two solutions of the i th equation we deduce that: a 1 +b 1 = λ 1 +ν 1 a +b = λ +ν a n +b n = λ n +ν n ut λ 1 +ν 1 λ +ν A+ = λ n +ν n and each diagonal entry of A+ is equal the sum of two nonegative eigenvalues of M, thus we have necessarily: M k A + k for all k = 1,,n which completes the proof Example 31 Let i x 99 i M x = 1 i 99 1 i 1 3 If 1 x 1, M x is positive definite and we have: for all symmetric norms, where A = M x A+ (8 ( ( x and = From Corollary?? if M x is positive definite for x = 3 1 then M x is PD for all x > 3 The eigenvalues of 1 9

11 M 3 1 which are the same as the singular values of M 3 are: 1 λ 1 = (9 λ = 19 + λ 3 = 149 λ 4 = 19 And the (8 inequality follows from Theorem ( ( (1 Let us study the commutation condition in Theorem 35 First notice that any square matrix X = (x ij M n will commute with A = diag(a 1,,a n if and only if : x 1,1 a 1 x 1, a x 1,n a n x 1,1 a 1 x 1, a 1 x 1,n a 1 x,1 a 1 x, a x,n a n = x,1 a x, a x,n a x n,1 a 1 x n, a x n,n a n x n,1 a n x n, a n x n,n a n Corollary 3 Let A = diag(λ 1,,λ n, = diag(ν 1,,ν n and M = X a given positive semi-definite matrix If X commute with A, or X commute with, then M A+ for all symmetric norms Proof As in Theorem 35, we will assume without loss of generality that X commute with A, as the other case is similar From the commutation condition we have Corollary 33 Let M = X be a positive semi-definite matrix written by blocks There exist a unitary V and a unitary U such that for all symmetric norms ( A X X UAU +VV := A + Proof Let U and V be two unitary matrix such that UAU = D o and VV = G o where D o and G o are two diagonal matrices having the same ordering o, of eigenvalues with respect to their indexes ie, if λ n λ 1 are the diagonal entries of D o, and ν n ν 1 are those of G o, then if λ i is in the (j,j position then ν i will be also 1

12 Consequently UAU +VV = D o +G o = D o + G o = A +, for all the Ky-Fan k norms and thus for all symmetric norms To complete the proof notice that ( U if T = UXV and Q is the unitary matrix, by Remark 1 V ( ( ( A X X = A X Q X Q = Do T T D G o + G o (13 o for all symmetric norms Lemma 3 Let N = a 1 a a n D D b 1 b b n where a 1,,a n respectively b 1,,b n are nonnegative respectively negative real numbers, A = a 1 a a n, = b 1 b b n and D is any diagonal matrix, then nor N neither N is positive semi-definite If a i +b i for all i n then N s > A+ s Proof The diagonal of N has negative and positive numbers, thus nor N neither N is positive semi-definite, now any two diagonal matrices will commute, in particular D and A, by applying Theorem 3 we get that the eigenvalues of N are the roots of det((a µi n ( µi n D D = Equivalently the eigenvalues are all the solutions of the following equations: (a 1 µ(b 1 µ d 1 = (a µ(b µ d = (a 3 µ(b 3 µ d 3 = (a n µ(b n µ d n = where (d 1,,d n are the diagonal entries of D D Let us denote by x i and y i the two solutions of the i th equation then: x 1 +y 1 = a 1 +b 1 x +y = a +b x n +y n = a n +b n 11

13 It seems easy to construct examples of non PSD matrices N, such that N s > A+ s, let us have a look of such inequality for PSD matrices Example 3 Let ( C = 5 A X 3 = 1 X ( 4 ( where A = 3 3, = Since the eigenvalues of C are all positive with 1 λ 1 38, λ 17, λ 3 9, λ 4 89, C is positive definite and we verify that 38 C s > A+ s = 3 Example 33 Let ( N y = y A X 1 = X ( ( 1 where A = and = The eigenvalues of N 1 y are the numbers: λ 1 = 4, λ = 1, λ 3 = y, λ 4 =, thus if y, N y is positive semi-definite and for all y such that y < 1 we have 1 4 = N y s > A+ s = 3 16+y +1 = N ( > A+ ( = 4(3+y+y +1 1

14 References [1] J C ourin, E Y Lee, and M Lin, On a decomposition lemma for positive semidefinite block-matrices, Linear Algebra and its Applications 437, pp , (1 [] J C ourin, F Hiai, Norm and anti-norm inequalities for positive semi-definite matrices, InternatJMath63, pp , (11 13