Block perturbation of symplectic matrices in Williamson’s theorem

Abstract Williamson’s theorem states that for any 
$2n \times 2n$
 real positive definite matrix A, there exists a 
$2n \times 2n$
 real symplectic matrix S such that 
$S^TAS=D \oplus D$
 , where D is an 
$n\times n$
 diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of A. Let H be any 
$2n \times 2n$
 real symmetric matrix such that the perturbed matrix 
$A+H$
 is also positive definite. In this paper, we show that any symplectic matrix 
$\tilde {S}$
 diagonalizing 
$A+H$
 in Williamson’s theorem is of the form 
$\tilde {S}=S Q+\mathcal {O}(\|H\|)$
 , where Q is a 
$2n \times 2n$
 real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that 
$\tilde {S}$
 and S can be chosen so that 
$\|\tilde {S}-S\|=\mathcal {O}(\|H\|)$
 . Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45–58, 2017].


I. INTRODUCTION
Analogous to the spectral theorem in linear algebra is Williamson's theorem [1] in symplectic linear algebra.It states that for any 2 × 2 real positive definite matrix , there exists a 2 × 2 real symplectic matrix  such that    =  ⊕ , where  is an  ×  diagonal matrix with positive diagonal entries.The diagonal entries of  are known as the symplectic eigenvalues of , and the columns of  form a symplectic eigenbasis of .This result is also referred to as Williamson normal form in the literature [2,3].Symplectic eigenvalues and symplectic matrices are ubiquitous in many areas such as classical Hamiltonian dynamics [4], quantum mechanics [3], and symplectic topology [5].More recently, it has attracted much attention from matrix analysts [6][7][8][9][10][11][12][13] and quantum physicists [14][15][16][17][18] for its important role in continuous-variable quantum information theory [19].For example, any Gaussian state of zero mean vector is obtained by applying to a tensor product of thermal states a unitary map that is characterized by a symplectic matrix [19], and the von-Neumann entropy of the Gaussian state is a smooth function of the symplectic eigenvalues of its covariance matrix [20].So, it is of theoretical interest as well as practical importance to study the perturbation of symplectic eigenvalues and symplectic matrices in Williamson's theorem, both of which are closely related to each other.Indeed, the perturbation bound on symplectic eigenvalues of two positive definite matrices  and  obtained in [11] is derived using symplectic matrices diagonalizing  + (1 − ) for  ∈ [0, 1].In [16], a perturbation of  of the form  +  was considered for small variable  > 0 and a fixed real symmetric matrix .The authors studied the stability of symplectic matrices diagonalizing  +  in Williamson's theorem and a perturbation bound was obtained for the case of  having non-repeated symplectic eigenvalues.
In this paper, we study the stability of symplectic matrices in Williamson's theorem diagonalizing  + , where  is an arbitrary 2 × 2 real symmetric matrix such that the perturbed matrix  +  is also positive definite.Let  be a fixed symplectic matrix diagonalizing  in Williamson's theorem.We show that any symplectic matrix S diagonalizing  +  in Williamson's theorem is of the form S =  + (∥ ∥) such that  is a 2 × 2 real symplectic as well as orthogonal matrix.Moreover,  is in symplectic block diagonal form with block sizes given by twice the multiplicities of the symplectic eigenvalues of .Consequently, we prove that S and  can be chosen so that ∥ S − ∥ = (∥ ∥).Our results hold even if  has repeated symplectic eigenvalues, generalizing the stability result of symplectic matrices corresponding to the case of non-repeated symplectic eigenvalues given in [16].We do not provide any perturbation bounds.
The rest of the paper is organized as follows.In Section II, we review some definitions, set notations, and define basic symplectic operations.In Section III, we detail the findings of this paper.These are given in Proposition 5, Theorem 6, Theorem 8, and Proposition 9.

II. BACKGROUND AND NOTATIONS
Let Sm() denote the set of  ×  real symmetric matrices equipped with the spectral norm ∥ • ∥, that is, for any  ∈ Sm(), ∥ ∥ is the maximum singular value of .We also use the same notation ∥ • ∥ for the Euclidean norm, and ⟨•, •⟩ for the Euclidean inner product on R  or C  .Let 0 , denote the  ×  zero matrix, and let 0  denote the  ×  zero matrix (i.e., 0  = 0 , ).We denote the imaginary unit number by  √ −1.We use the Big-O notation  = (∥ ∥) to denote a matrix  as a function of  for which there exist positive scalars  and  such that ∥∥ ≤  ∥ ∥ for all  with ∥ ∥ < .

A. Symplectic matrices and symplectic eigenvalues
Define  2 0 1 −1 0 , and let  2 =  2 ⊗   for  > 1, where   is the  ×  identity matrix.A 2 × 2 real matrix  is said to be symplectic if    2  =  2 .The set of 2 × 2 symplectic matrices, denote by Sp(2), forms a group under multiplication called the symplectic group.The symplectic group Sp(2) is analogous to the orthogonal group Or(2) of 2 × 2 orthogonal matrices in the sense that replacing the matrix  2 with  2 in the definition of symplectic matrices gives the definition of orthogonal matrices.However, in contrast with the orthogonal group, the symplectic group is non-compact.Also, the determinant of every symplectic matrix is equal to +1 which makes the symplectic group a subgroup of the special linear group [3].Let Pd(2) ⊂ Sm(2) denote the set of positive definite matrices.Williamson's theorem [1] states that for every  ∈ Pd(2), there exists  ∈ Sp(2) such that    =  ⊕ , where  is an  ×  diagonal matrix.The diagonal elements  1 () ≤ • • • ≤   () of  are independent of the choice of , and they are known as the symplectic eigenvalues of .Denote by Sp(2; ) the subset of Sp(2) consisting of symplectic matrices that diagonalize  in Williamson's theorem.Several proofs of Williamson's theorem are available using basic linear algebra (e.g., [2,21]).

B. Symplectic block and symplectic direct sum
Let  be a natural number and ℐ,  ⊆ {1, . . ., }.Suppose  is an  ×  matrix.We denote by   the submatrix of  consisting of the columns of  with indices in .Also, denote by Also, define a symplectic diagonal block of  as a submatrix of the form The following example illustrates this. .
Let  ′ be another 2 ′ × 2 ′ matrix, given in the block form where the blocks  ′ ,  ′ ,  ′ ,  ′ have size  ′ ×  ′ .Define the symplectic direct sum of  and  ′ as This is illustrated in the following example.,  ′ = 17 18 19 20 .
We then have We know that the usual direct sum of two orthogonal matrices is also an orthogonal matrix.It is interesting to note that an analogous property is also satisfied by the symplectic direct sum.If  ∈ Sp(2) and  ′ ∈ Sp(2ℓ ), then  ⊕   ′ ∈ Sp(2( + ℓ )).Indeed, we have

III. MAIN RESULTS
We fix the following notations throughout the paper.Let  ∈ Pd(2) with distinct symplectic eigenvalues  1 < • • • <   .For all  = 1, . . ., , define sets An example to illustrate these sets is as follows.
The matrix  in Theorem 6 characterizes the set Sp(2; ).We state this in the following proposition, proof of which follows directly from Corollary 5.3 of [11].It is also stated as Theorem 3.5 in [22].Proposition 7. Let  ∈ Sp(2; ) be fixed.Every symplectic matrix Ŝ ∈ Sp(2; ) is precisely of the form In [16], it is shown that if  has no repeated symplectic eigenvalues, then for any fixed  ∈ Sm(2), one can choose  ∈ Sp(2; ) and () ∈ Sp(2;  + ) for small  > 0 such that ∥() − ∥ = ( √ ).We generalize their result to the more general case of  having repeated symplectic eigenvalues.Moreover, we consider the most general perturbation of  and strengthen the aforementioned result.(25) By ( 8) and (28) we have which implies Range( [+1] ) is non-isotropic for small (∥ ∥).Apply ESR to and From ( 8) and (30), we get Combining ( 30) and (32) then gives We thus have To complete the induction, we just need to show that )).We have By the necessary and sufficient condition for ( [1] Substitute in (33) the value of  [+1] from (29) to get Apply the induction hypothesis ) and simplify as follows: We have thus shown that
consisting of the elements   with indices  ∈ ℐ and  ∈ .Let  be any 2 × 2 matrix given in the block form by  =     , where  , ,  ,  are matrices of order  × .Define a symplectic block of  as a submatrix of the form

.
The symplectic concatenation of  and  is given by  ⋄  =