Some inequalities for some positive block matrices

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Some inequalities for some positive block matrices

Antoine Mhanna

To cite this version:

Antoine Mhanna. Some inequalities for some positive block matrices. 2021. �hal-03171502�

Some inequalities for some positive block matrices

Antoine Mhanna ¹

Kfardebian, Lebanon

Abstract

We review Lin’s inequality from [2] and (re)prove that if M = [X i,j ] ⁿ _i,j=1 is positive semi-definite then L n

i=1 (X _i,i − P

j6=i

X _j,j ) ≤ M ^τ ≤ I _n ⊗

n

P

i=1

X _i,i where M ^τ is the partial transpose of M. In particular for such M ∈ M ⁺ n ( M m ) we prove that kM k ≤ min(m, n)k P ⁿ

i=1

X _i,i k for all symmetric norms. Some classical results are also discussed in terms of permutations.

Keywords: positive block matrix; partial transpose; symmetric norm;

permutations.

2010 MSC: 15A60; 15A42; 15B99; 05A18.

1. Introduction and preliminaries

Let M n (M m ) denote the space of n × n complex matrices with entries in M m ( C ). We write M ^H n ( M m ) respectively M ⁺ n ( M m ) for the set of Hermitian block matrices respectively positive semi-definite matrices in M n ( M m ). Let Im(X) := X − X ^∗

2i resp. Re(X) := X + X ^∗

2 be the imaginary part resp.

the real part of a matrix X, if W (X) denotes the field of values of X then W (Re(X)) = <(W (X)) and W (Im(X)) = =(W (X)) see [3]. A positive semi-definite matrix A is denoted by A ≥ 0 and a norm k.k over the space of matrices is a symmetric norm if kU AV k = kAk for all A and all unitaries U and V.

For a matrix M = [X i,j ] ∈ M n ( M m ) we denote M ^τ the matrix [X _i,j ^∗ ] where each block is replaced by its adjoint. We define Tr ₁ (M ) := P n

i=1 X _i,i , Tr 2 (M ) := [Tr(X i,j )] for Tr 1 (M ) resp. Tr 2 (M ) is an m × m matrix resp. an n × n complex matrix. M is positive partial transpose (P.P.T.) if M ≥ 0

1

Email: tmhanat@yahoo.com

and M ^τ ≥ 0. I m (or I) is the identity matrix of dimension m and a 0 entry is an all zero submatrix. We finally denote λ _i (M) resp. σ _i (M ), i = 1, . . . , n the eigenvalues resp. the singular values of M in decreasing order.

We need the following notations used latter in the article:

For X ∈ M n , the vertical width ω _v (X) respectively the width ω(X) of W (X) is the smallest possible real number such that W (X) is contained in a vertical strip of width ω v (X) respectively in a strip of width ω(X) with ω(X) ≤ ω _v (X).

If S = {M _k , 1 ≤ k ≤ i, M _k ∈ M m } of cardinality i, {S} denotes the 2 ⁱ⁻¹ set of matrices {M ₁ ± . . . ± M i } and an element in {S} is denoted by S ^[σ] , we write ω _v ({S}) to mean the 2 ⁱ⁻¹ tuple ω _v (S ^[σ] ). Notice that if the width of W (A) is ω, then one can find a θ ∈ [0, π] such that

rI _m ≤ Re(e ^iθ A) ≤ (r + ω)I _m for some r ∈ R and ω = ω v (e ^iθ A).

Finally if M := [X _i,j ] ⁿ _i,j=1 ∈ M n ( M m ) with blocks X _i,j we write M = [

[x _k,l ] _i,j ] where

[x _k,l ] _i,j is the complex number belonging to block X _i,j on its line k and column l for k, l: 1 . . . m and i, j: 1 . . . n one has

M T

[x _k,l ] _i,j =

[x _l,k ] _j,i ;

[x _k,l ] _i,j =

[x _l,k ] _i,j . The matrix ˜ M defined in M m ( M n ) by

M

[x _i,j ] _k,l =

[x _k,l ] _i,j is known to be permutationally congruent to M i.e there exist a permutation P _M such that P _M ^T MP _M = ˜ M, see [10, Theorem 7] and [12, Theorem 2.4] with ( ˜ ] M ) ^τ = (M ^τ ) ^T .

We sketch P _M as follows: An index in [1, mn] is denoted by ι, upon completing M by m × m zero blocks or extending each block by 0 ⁰ s we consider the completed matrix M ₀ ∈ M r ( M r ) with max(m, n) = r, take the transposition T that permutes: for every s, 0 ≤ s ≤ r − 2 and sr + 2 + s ≤ ι ≤ sr + r, ι ↔ (ι − sr − 1)r + 1 + s, T M 0 T = M f 0 . Applying block swapping (Proposition 1.1) one can show that there is some permutation P such that

P ^T

M 0 0 0

P =

M ˜ 0 0 0

.

Assume that M is invertible then so is ˜ M , writing P by blocks P = L ?

? ?

we see that L ^T M L = M ˜ which implies that L is invertible and thus a permutation matrix. By a continuity argument we can take P _M = L, see [14] and the references therein for further reading.

2

Proposition 1.1. Let A = [A i,j ] ∈ M s ( M p ) and let B = [B i,j ] ∈ M s ( M q ) two complex matrices written by blocks. For C = [C _i,j ] ∈ M s ( M p+q ) where C i,j =

A i,j 0 0 B i,j

there is a permutation P such that P ^T CP =

A 0

0 B

.

Proof. A (u, v, r) block swap B _uvr is a permutation matrix (here of dimen- sion s(p + q)) of the form

I

0 0 0 0 0 I

0 0 I

0 0 0 0 0 I

!

. Say s ≥ 2 for i = 1 . . . s − 1, take P = Q s−1

i=1 B u

v

r

where u i = ip, v i = iq and r i = p.

2. Main results If M =

A X X ^∗ B

∈ M ^H 2 ( M m ) (not necessarily ≥ 0) with positive semi- definite diagonal blocks then it is easy to see that |λ _n (M)| ≤ λ 1 (M) (to see this consider the eigenvalues of

0 X X ^∗ 0

). In particular if M ≤ I ⊗ S then kMk _s ≤ kSk _s where k.k _s is the spectral norm. This is not true for n > 2 with n = 4 taking for example









0 3 −1 0

3 0 2 0

−1 2 0 1

0 0 1 0









where λ 4 ≈ −4.1 and λ ₁ ≈ 3.22. In fact if M is an entry wise nonnegative matrix we know by Perron-Frobenius theorem that |λ _n (M )| ≤ λ 1 (M).

2.1. Lin’s Inequality

In [1] the following norm inequality is proved:

Theorem 2.1. [1] Let M =

A X X ^∗ B

∈ M ⁺ ₂ (M m ). Then for all symmetric norms

kMk ≤ kA + B + ωI m k,

where ω is the width of a strip containing the numerical range of X.

The previous theorem has been generalized in [4] for 2×2 block matrices.

In [2] a generalization to any number of blocks was presented, here we refine Lin’s idea mainly from Theorem’s 2.1 proof (the base case): As M is positive,

we may write M =









 X ₁ ^∗ X ₂ ^∗ .. . X _n ^∗











X 1 X 2 . . . X n

for some X i ∈ M mn×m so

that M i,i = X _i ^∗ X i and kM k _k = k

n

X

i=1

X i X _i ^∗ k _k (k.k _k are Ky-Fan k-norms).

Identifying the n-tuple X to the set {X _i , 1 ≤ i ≤ n} we have (by a direct induction) P n

i=1 X i X _i ^∗ = 1 2 ⁿ⁻¹ ( P

σ (X ^[σ] )(X ^[σ] ) ^∗ ), applying Ky-Fan k-norms triangular inequality we get:

kM k _k = 1 2 ⁿ⁻¹ k X

σ

(X ^[σ] )(X ^[σ] ) ^∗ k _k ≤ 1 2 ⁿ⁻¹

X

σ

k(X ^[σ] ) ^∗ (X ^[σ] )k _k (2.1) Theorem 2.2. [2] Let M = [X i,j ] ∈ M ⁺ n ( M m ) and let S i = {X _i,k , i + 1 ≤ k ≤ n} for i = 1, . . . , n − 1 then for all symmetric norms

kM k ≤ k

n

X

i=1

X i,i +

n−1

X

i=1

υ i I m k,

where υ i is the 2 ⁿ⁻ⁱ⁻¹ average of ω v ({S _i }).

Proof. The proof follows from [2] or one can use (2.1), each term in the σ sum is a positive semi-definite matrix that can be written as:

n

X

i=1

X _i ^∗ X _i +

n−1

X

i=1

±Re(X _i ^∗ (±X _i+1 · · · ± X _n )),

by Ky-Fan dominance theorem we can (after bounding the vertical width of each matrix in {S _i } by a scalar identity) split the quantities to get the result.

Corollary 2.3. Let M = [X _i,j ] ∈ M ⁺ _n ( M m ), then for all symmetric norms kM k ≤ k

n

X

i=1

X _i,i +

n

X

i=2

υ _i I _m k,

where υ _i (i = 2, . . . , n) is the 2 ⁱ⁻² average of ω _v ({S _i }) and S _i = {X _k,i , 1 ≤ k ≤ i − 1}.

Proof. Consider the matrix LM L in Theorem 2.2 with L the anti-diagonal permutation matrix for blocks (L =

_{0 ··· I}

.. . ... ...

I

··· 0

).

Corollary 2.4. Let M = [X _i,j ] ∈ M ⁺ n ( M m ) such that for i < j, r _i,j I _m ≤ Re(X i,j ) ≤ r i,j I m + δ i,j I m then kM k ≤ k

n

P

i=1

X i,i + P

i<j δ i,j I m k for all symmetric norms.

4

Proof. The proof follows from Theorem 2.2 by writing each X i,j = Re(X i,j )+

iIm(X _i,j ) for i < j in the average expression and bounding the real parts we get the stated inequality.

The case Re(X _i,j ) = 0.I _m (skew-Hermitian blocks) is the main result of [15]. We point out also that in Theorem 2.1 of [2] the author replaced (by a miss-justification when n > 2 as per our private communication) vertical widths by smallest widths ω and proposed the same replacement in Corollary 2.4 as Question 2.5.

2.2. Around Hiroshima majorization

In this section we reprove some classical results concerning matrices in M ^H n ( M m ) related to some permutations and P.P.T. matrices, the approach is close (dual) to that in [9]. Despite known this let us prove some non trivial inequalities in the last subsection.

Corollary 2.5. [9] Let M ∈ M ^H n ( M m ) such that M ^τ ≤ Tr ₂ (M ^τ ) ⊗ I _m then there exist a permutation P _M with:

N = P _M ^T M ^τ P _M ≤ P _M ^T (Tr ₂ (M ^τ ) ⊗ I _m )P _M = I _m ⊗ Tr ₁ (N ) (2.2) for N ∈ M ^H m ( M n ), Tr ₁ (N ) = Tr ₂ (M ^τ ) is an n × n matrix. Furthermore if M ≥ 0 then N ^τ ≥ 0.

Remark 2.6. Set M ∈ M n ( M m ), U unitary and U ^∗ M U ∈ M n ( M m ). Define the block m × m permutation matrix P M ∈ M n (M m ) having zero or identity m × m blocks; if U = P M is a transposition block matrix then it is easy to see that (P _M M P _M ) ^τ = P _M M ^τ P _M and thus upon composition for any block m × m permutation matrix P M , P M (P _M ^T M P M ) ^τ P _M ^T = M ^τ .

Let M ∈ M ^H n ( M m ) such that M ^τ ≥ 0 and U unitary, for U ^∗ M U ∈ M ^H _n (M m ) the matrix (U ^∗ M U ) ^τ may not be positive semi-definite in general;

consider M = U N U ^∗ ∈ M ⁺ 3 ( M 2 ) in Example 2.12 with U =





0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1



,

however this is true if n = m and U = P _M , this is also true for M ∈ M ^H ₂ (M m ) and U ^∗ =

e ^iζ cos(θ)I sin(θ)I

−e ^iζ sin(θ)I cos(θ)I

([4]).

From Quantum Physics theory we know that if M is P.P.T. then M ≤

I _n ⊗ Tr ₁ (M) and M ≤ Tr ₂ (M ) ⊗ I _m see for example [6, 7] and more recently

[8, 9]. Here we present a different proof as follows:

Lemma 2.7. If M = [X i,j ] ∈ M ⁺ n ( M m ) then

n

M

i=1

(X _i,i − X

j6=i

X _j,j ) ≤ M ^τ ≤ I _n ⊗

n

X

i=1

X _i,i .

Proof. We prove the right side inequality as the lower bound has a similar proof by noticing that

A X X ^∗ B

≥ 0 ⇐⇒

A −X

−X ^∗ B

≥ 0. We write E = I _n ⊗ P ⁿ

i=1

X _i,i − M ^τ as the sum of positive semi-definite matrices of the form E _k,j =

X _j,j (k) −X _k,j ^∗

−X _k,j X _k,k (j)

where we assumed E _k,j (k < j) is an mn × mn block matrix having all zero blocks except the −X _k,j (−X _k,j ^∗ ) blocks taken on the same position as in E; with X _j,j (k) to mean X _j,j on the k ^th diagonal entry of E. Since M ≥ 0, up to the congruence

0 −I I 0

each E k,j is a principal block submatrix of M and E = P

k<j E k,j ≥ 0.

Remark 2.8. Let M = [X i,j ] ∈ M ⁺ n ( M m ) one can check following the previous proof that M ≤ L n

i=1 nX _i,i .

The last lemma and corollary imply the following:

Corollary 2.9. [9] If M = [X i,j ] ∈ M ⁺ n ( M m ) then M ^τ ≤ Tr 2 (M ^τ ) ⊗ I m . In [5]-Theorem 1- the following (Hiroshima majorization) is proved:

Theorem 2.10. [5] Let M = [X i,j ] ∈ M ⁺ _n (M m ) such that M ≤ Tr 2 (M ) ⊗I _m then for all k ∈ [1, mn] :

k

X

i=1

λ i (M ) ≤

k

X

i=1

λ i (Tr 2 (M)).

Corollary 2.11. Let M = [X i,j ] ∈ M ^H n ( M m ) such that 0 ≤ M ≤ I _n ⊗

n

X

i=1

X _i,i (2.3)

then for all k ∈ [1, m] :

k

X

i=1

σ _i (M ) ≤

k

X

i=1

σ _i (

n

X

i=1

X _i,i )

equivalently kM k ≤ k P ⁿ

i=1

X _i,i k for all symmetric norms.

6

Upon completing

n

P

i=1

X i,i by zeros we can take k; 1 ≤ k ≤ mn to get the last inequality from Ky-Fan dominance theorem ([3] -Sec 10.7-). In [13] an- other direct proof of Corollary 2.11 is given with the fact that Corollary 2.11 and Theorem 2.10 are equivalent (see Corollary 2.5).

Example 2.12. It is well known that a matrix in M ⁺ n ( M m ) (n > 2) satisfy- ing (2.3) need not be P.P.T. as for example N ∈ M ⁺ ₃ ( M 2 ) =





4 0 0 2 0 0 0 4 0 0 1 0 0 0 1 0 0 0 2 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0



 .

2.3. Consequences and Inequalities

The next corollary can be proved from the double inequality of Lemma 2.7.

Corollary 2.13. If M ∈ M ⁺ n ( M m ) then kM ^τ k _s ≤ kTr ₁ (M )k _s .

The positive condition in Corollary 2.11 seems necessary as this example shows:

Example 2.14. M =









1 0 0 2

0 4.5 0 0

0 0 0.9 0

2 0 0 1









∈ M 2 (M 2 ) and σ 1 (M ) + σ 2 (M ) = 4.5 + 3 > 7.4

For a matrix D = [D _i,j ] ∈ M ⁺ n ( M m ) with all blocks diagonal, it is easily seen that D is P.P.T. as D ^τ = ¯ D ( ¯ D is D with conjugate entries), upon considering the rearrangement of D taking P _D ^T DP _D we see that each di- agonal entry of Tr ₁ (D) is the sum of n eigenvalues of D which implies:

kDk ≤ kTr ₁ (D)k for all symmetric norms.

Corollary 2.15. [11] Let M = [X _i,j ] ∈ M ⁺ n ( M m ) then M ≤ mTr ₂ (M ) ⊗I _m . Proof. Take M P

= P _M ^T MP _M = [X _i,j ⁰ ] ∈ M ⁺ m ( M n ), applying Remark 2.8 M P

≤ L m

i=1 mX _i,i ⁰ , reversing the rearrangement M ≤ mD where D = [D i,j ] ∈ M ⁺ _n (M m ) and D i,j is the (main) diagonal of X i,j . Since D is P.P.T.

we get the result from Corollary 2.9.

Keeping the notations of previous corollary we have kMk ≤ mkTr ₁ (D)k for all symmetric norms, by the variational representation of Ky-Fan norms and Ky-Fan dominance Theorem (see [3]) Remark 2.8 gives:

Corollary 2.16. For M = [X i,j ] ∈ M ⁺ n ( M m ), kM k ≤ min(m, n)kTr ₁ (M )k

for all symmetric norms.

Acknowledgments

I thank Prof. Minghua Lin for correspondences.

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