Weak supermajorization between symplectic spectra of positive definite matrix and its pinching

W eak supermajorization betw een sym plectic spectr a of positiv e definite matrix and its pinching Temjensangba * and Hemant K. Mishra † Abstract. Let 𝐴 = h 𝐸 𝐹 𝐹 𝑇 𝐺 i be a 2 𝑛 × 2 𝑛 real positive definite matrix, where 𝐸 , 𝐹 , and 𝐺 are 𝑛 × 𝑛 blocks. It is shown that 𝑑 ( 𝐸 ⊕ 𝐺 ) ≺ 𝑤 𝑑 ( 𝐴 ) . Here 𝑑 ( 𝐴 ) denotes the 𝑛 -vector consisting of the symplectic eigenvalues of 𝐴 arranged in the non-decreasing order. W e also observe the fo llowing weak supermajorization relation, which is interesting on its own: 𝜆   𝒞 ( 𝐺 ) 1 / 2 𝒞 ( 𝐸 ) 𝒞 ( 𝐺 ) 1 / 2  1 / 2  ≺ 𝑤 𝜆   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  . Here 𝜆   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  denotes the 𝑛 -vector with entries given by the eigenvalues of  𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2 in the non-decreasing order. 1 Introduction Symplectic eigenvalues have been extensively explored over the last two decades and it continues to be a vibrant research field. Several analogs of classic eigenvalue results are developed for symplectic eigenval- ues [ 1 – 8 ]. In particular, symplectic versions of several majorization results on eigenvalues, such as Lidskii’s theorem [ 4 ] and Schur–Horn theorem [ 9 , 10 ] are known today, and these results work with weak super- majorization in place of majorization. For instance, it is known that the eigenvalues of a Hermitian matrix majorize the eigenvalues of its pinching (see, e.g., [ 11 ]). A symplectic analog of this result given in [ 12 , eorem 9] states that the symplectic eigenvalues of a positive definite matrix weakly supermajorize the symplectic eigenvalues of its symplectic pinching . It is to be noted that a symplectic pinching operation is fun- damentally different from a classical pinching operation. To the best of our knowledge, no symplectic analog of the aforementioned eigenvalue result is known for a classical pinching. In this letter, we establish a weak supermajorization relation between the symplectic eigenvalues of a real positive definite matrix and that of its pinching leaving two diagonal blocks of the same size. More concretely, let 𝐴 =  𝐸 𝐹 𝐹 𝑇 𝐺  be a 2 𝑛 × 2 𝑛 real positive definite matrix, where 𝐸 , 𝐹 , and 𝐺 are 𝑛 × 𝑛 matrices. In our main result, we show that the symplectic eigenvalues of 𝐸 ⊕ 𝐺 are weakly supermajorized by the symplectic eigenvalues of 𝐴 . We also report some interesting consequences of the main result. e rest of the paper is structured as follows. In Section 2 , we briefly discuss some prerequisites. e main result and its consequences are given in Section 3 . 2 Preliminaries In this section, we briefly discuss the prerequisites needed for our discourse. Denote by M 𝑚, 𝑛 the set of all 𝑚 × 𝑛 real matrices and let M 𝑛 : = M 𝑛, 𝑛 . Let P 𝑛 ⊂ M 𝑛 consisting of the positive definite matrices. For 𝐴 ∈ M 𝑛 , let 𝜎 ( 𝐴 ) denote the eigen spectrum of 𝐴 . If all the eigenvalues of 𝐴 are real, let 𝜆 ( 𝐴 ) = [ 𝜆 1 ( 𝐴 ) , . . . , 𝜆 𝑛 ( 𝐴 ) ] 𝑇 K eywords: Positive definite matrix, pinching, symplectic eigen value, weak supermajorization. MSC: 15B48, 15A18, 15A42. * Department of Mathematics and Computing, Indian Institute of Technology (ISM) Dhanbad, Jharkhand 826004, India; temjensangba111@gmail.com † Department of Mathematics and Computing, Indian Institute of Technology (ISM) Dhanbad, Jharkhand 826004, India; hemantmishra1124@iitism.ac.in 2026/03/31 01:08 Weak supermajorization between symplectic spectra of positive denite matrix and its pinching 2 denote the 𝑛 -vector whose entries are the usual eigenvalues of 𝐴 arranged in the non-decreasing order. For any 𝐴 = [ 𝑎 𝑖 𝑗 ] ∈ M 𝑛 , we write Δ ( 𝐴 ) = [ 𝑎 11 , . . . , 𝑎 𝑛𝑛 ] 𝑇 . A matrix 𝑀 ∈ M 𝑛 is said to be symplectic if it satisfies 𝑀 𝑇 𝐽 2 𝑛 𝑀 = 𝐽 2 𝑛 , (2.1) where 𝐽 2 𝑛 : =  0 𝐼 𝑛 − 𝐼 𝑛 0  , and 𝐼 𝑛 denotes the identity matrix of size 𝑛 × 𝑛 . We shall drop the “ 2 𝑛 ” from 𝐽 2 𝑛 wherever it is clear from the context. e set of all 2 𝑛 × 2 𝑛 symplectic matrices forms a group under matrix multiplication called the symplectic group, and we denote this group by SP 2 𝑛 . A set of vectors { 𝑥 1 , . . . 𝑥 𝑛 , 𝑦 1 , . . . , 𝑦 𝑛 } in R 2 𝑛 is said to be a symplectic basis if it satisfies for each 1 ≤ 𝑘 , ℓ ≤ 𝑛 : 𝑥 𝑇 𝑘 𝐽 𝑥 ℓ = 𝑦 𝑇 𝑘 𝐽 𝑦 ℓ = 0 , (2.2) 𝑥 𝑇 𝑘 𝐽 𝑦 ℓ = ( 1 , if 𝑘 = ℓ 0 , if 𝑘 ≠ ℓ . (2.3) ere is a one-to-one correspondence between the set of all symplectic bases of R 2 𝑛 and the symplectic group SP 2 𝑛 . Williamson’s theorem [ 13 ] states that for any 𝐴 ∈ P 2 𝑛 , there exists 𝑀 ∈ SP 2 𝑛 such that 𝑀 𝑇 𝐴 𝑀 = 𝐷 ⊕ 𝐷 , (2.4) where 𝐷 ∈ P 𝑛 is a diagonal matrix which is unique up to permutation of its diagonal entries. e diagonal entries of 𝐷 are called the symplectic eigenvalues of 𝐴 . Let 𝜎 𝑠 ( 𝐴 ) denote the set of all symplectic eigenvalues of 𝐴 and call it the symplectic spectrum of 𝐴 . Let 𝑑 ( 𝐴 ) = [ 𝑑 1 ( 𝐴 ) , . . . , 𝑑 𝑛 ( 𝐴 ) ] 𝑇 denote the 𝑛 -vector con- sisting of the symplectic eigenvalues of 𝐴 with entries arranged in the non-decreasing order. e set of all symplectic matrices which diagonalizes 𝐴 in the sense of Williamson’s theorem ( 2.4 ) is denoted by SP 2 𝑛 ( 𝐴 ) . If { 𝑥 1 , . . . , 𝑥 𝑛 , 𝑦 1 , . . . , 𝑦 𝑛 } is the symplectic basis of R 2 𝑛 given by the columns of 𝑀 in ( 2.4 ), then we have 𝐴𝑥 𝑘 = 𝑑 𝑘 ( 𝐴 ) 𝐽 𝑦 𝑘 , 𝐴 𝑦 𝑘 = − 𝑑 𝑘 ( 𝐴 ) 𝐽 𝑥 𝑘 , ∀ 1 ≤ 𝑘 ≤ 𝑛 . (2.5) We call each pair ( 𝑥 𝑘 , 𝑦 𝑘 ) a symplectic eigenvector pair of 𝐴 corresponding to the symplectic eigenvalue 𝑑 𝑘 ( 𝐴 ) . It follows from ( 2.1 ) and ( 2.4 ) that 𝑀 ∈ SP 2 𝑛 ( 𝐴 ) if and only if the 𝑘 th and the ( 𝑛 + 𝑘 ) th columns of 𝑀 form a symplectic eigenvector pair of 𝐴 corresponding to the symplectic eigenvalue 𝑑 𝑘 ( 𝐴 ) for all 𝑘 ∈ { 1 , . . . , 𝑛 } . Suppose 𝑚 1 , . . . , 𝑚 𝑘 ∈ N and 𝑛 = 𝑚 1 + . . . + 𝑚 𝑘 . Let 𝐻 = [ 𝐻 𝑖 𝑗 ] ∈ M 𝑛 with blocks 𝐻 𝑖 𝑗 ∈ M 𝑚 𝑖 , 𝑚 𝑗 for 1 ≤ 𝑖, 𝑗 ≤ 𝑘 . A pinching of 𝑀 relative to ( 𝑚 1 , . . . , 𝑚 𝑘 ) , denoted by 𝒞 ( 𝐻 ) , is defined as 𝒞 ( 𝐻 ) = 𝑘 Ê 𝑗 = 1 𝐻 𝑗 𝑗 , (2.6) which is the direct sum of 𝐻 11 , . . . , 𝐻 𝑘 𝑘 . Let 𝐴 =  𝐸 𝐹 𝐹 𝑇 𝐺  ∈ P 2 𝑛 , where 𝐸 , 𝐹 , 𝐺 are 𝑛 × 𝑛 blocks. e symplectic pinching or 𝑠 -pinching of 𝐴 relative to ( 𝑚 1 , . . . , 𝑚 𝑘 ) , denoted by 𝒞 𝑠 ( 𝐴 ) , is defined as 𝒞 𝑠 ( 𝐴 ) : =  𝒞 ( 𝐸 ) 𝒞 ( 𝐹 ) 𝒞 ( 𝐹 𝑇 ) 𝒞 ( 𝐺 )  . (2.7) Also, define Δ 𝑠 ( 𝐴 ) : =  √ 𝜂 1 𝛾 1 , . . . , √ 𝜂 𝑛 𝛾 𝑛  𝑇 , (2.8) where 𝜂 𝑖 and 𝛾 𝑖 are the 𝑖 th diagonal entries of 𝐸 and 𝐺 , respectively, for 1 ≤ 𝑖 ≤ 𝑛 . 2026/03/31 01:08 Weak supermajorization between symplectic spectra of positive denite matrix and its pinching 3 For any 𝑥 ∈ R 𝑛 , we denote the entries of 𝑥 arranged in non-decreasing order by 𝑥 ↑ 1 , . . . , 𝑥 ↑ 𝑛 . For any 𝑥 , 𝑦 ∈ R 𝑛 , we say that 𝑥 is weakly supermajorized by 𝑦 , denoted by 𝑥 ≺ 𝑤 𝑦 , if 𝑘  𝑗 = 1 𝑥 ↑ 𝑗 ≥ 𝑘  𝑗 = 1 𝑦 ↑ 𝑗 , 1 ≤ 𝑘 ≤ 𝑛 . (2.9) If equality holds in ( 2.9 ) for 𝑘 = 𝑛 , we say that 𝑥 is majorized by 𝑦 , denoted by 𝑥 ≺ 𝑦 . 3 Main result roughout this section, we shall consider 𝐴 =  𝐸 𝐹 𝐹 𝑇 𝐺  ∈ P 2 𝑛 , where 𝐸 , 𝐹 , 𝐺 are 𝑛 × 𝑛 blocks. It will be worthwhile to recall from [ 9 , Section 2] that, if 𝐸 , 𝐹 , and 𝐺 are diagonal matrices, then 𝜎 𝑠 ( 𝐴 ) =   𝜂 1 𝛾 1 − 𝛽 2 1 , . . . ,  𝜂 𝑛 𝛾 𝑛 − 𝛽 2 𝑛  , (3.1) where 𝜂 𝑗 , 𝛾 𝑗 , and 𝛽 𝑗 are the 𝑗 th diagonal entries of 𝐸 , 𝐺 , and 𝐹 for 1 ≤ 𝑗 ≤ 𝑛 , respectively. e following lemma will be useful in the proof of the main result. Lemma 3.1 We have 𝑑 ( 𝐸 ⊕ 𝐺 ) = 𝜆   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  . (3.2) Consequently, we have 𝜎 𝑠 ( 𝐸 ⊕ 𝐺 ) = n √ 𝜆 : 𝜆 ∈ 𝜎 ( 𝐸 𝐺 ) o . (3.3) Proof: We have  𝐺 1 / 2 ⊕ 𝐺 − 1 / 2  ( 𝐸 ⊕ 𝐺 )  𝐺 1 / 2 ⊕ 𝐺 − 1 / 2  = ( 𝐺 1 / 2 𝐸 𝐺 1 / 2 ) ⊕ 𝐼 𝑛 . (3.4) By the spectral theorem, there exists an orthogonal matrix 𝑃 such that 𝐺 1 / 2 𝐸 𝐺 1 / 2 = 𝑃 𝑇 Λ 𝑃 , where Λ = diag ( 𝜆 ( 𝐺 1 / 2 𝐸 𝐺 1 / 2 ) ) . Since  𝐺 1 / 2 ⊕ 𝐺 − 1 / 2  is a symplectic matrix, it follows that 𝑑 ( 𝐸 ⊕ 𝐺 ) = 𝑑   𝐺 1 / 2 𝐸 𝐺 1 / 2  ⊕ 𝐼 𝑛  (3.5) = 𝑑  𝑃 𝑇 Λ 𝑃 ⊕ 𝐼 𝑛  (3.6) = 𝑑  ( 𝑃 ⊕ 𝑃 ) 𝑇 ( Λ ⊕ 𝐼 𝑛 ) ( 𝑃 ⊕ 𝑃 )  (3.7) = 𝑑 ( Λ ⊕ 𝐼 𝑛 ) (3.8) =   𝜆 1  𝐺 1 / 2 𝐸 𝐺 1 / 2  , . . . ,  𝜆 𝑛  𝐺 1 / 2 𝐸 𝐺 1 / 2   𝑇 (3.9) =  𝜆 1   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  , . . . , 𝜆 𝑛   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2   𝑇 , (3.10) where ( 3.9 ) follows from ( 3.1 ). is proves ( 3.2 ). Since 𝐸 𝐺 = 𝐺 − 1 / 2  𝐺 1 / 2 𝐸 𝐺 1 / 2  𝐺 1 / 2 , we have 𝜎 ( 𝐸 𝐺 ) = 𝜎  𝐺 1 / 2 𝐸 𝐺 1 / 2  , (3.11) 2026/03/31 01:08 Weak supermajorization between symplectic spectra of positive denite matrix and its pinching 4 which implies that n √ 𝜆 : 𝜆 ∈ 𝜎 ( 𝐸 𝐺 ) o = 𝜎   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  . (3.12) e relations ( 3.2 ) and ( 3.12 ) imply ( 3.3 ). ■ e following theorem is the main result. Theorem 3.2 We have 𝑑 ( 𝐸 ⊕ 𝐺 ) ≺ 𝑤 𝑑 ( 𝐴 ) . (3.13) Proof: We have seen in ( 3.11 ) that 𝜎 ( 𝐸 𝐺 ) = 𝜎  𝐺 1 / 2 𝐸 𝐺 1 / 2  . (3.14) By the spectral theorem, there exists an orthonormal basis { 𝑣 1 , . . . , 𝑣 𝑛 } of R 𝑛 such that 𝐺 1 / 2 𝐸 𝐺 1 / 2 𝑣 𝑘 = 𝜆 𝑘 ( 𝐸 𝐺 ) 𝑣 𝑘 , ∀ 1 ≤ 𝑘 ≤ 𝑛 . (3.15) Now, define 𝑢 𝑘 : =  𝑑 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝐺 − 1 / 2 𝑣 𝑘 . Observe that 𝐸 𝐺 𝑢 𝑘 = 𝐸 𝐺   𝑑 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝐺 − 1 / 2 𝑣 𝑘  (3.16) =  𝑑 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝐸 𝐺 1 / 2 𝑣 𝑘 (3.17) =  𝑑 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝐺 − 1 / 2  𝐺 1 / 2 𝐸 𝐺 1 / 2 𝑣 𝑘  (3.18) = 𝜆 𝑘 ( 𝐸 𝐺 )   𝑑 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝐺 − 1 / 2 𝑣 𝑘  (3.19) = 𝜆 𝑘 ( 𝐸 𝐺 ) 𝑢 𝑘 . (3.20) By substituting 𝜆 𝑘 ( 𝐸 𝐺 ) = 𝑑 2 𝑘 ( 𝐸 ⊕ 𝐺 ) from ( 3.3 ) into ( 3.20 ), we get 𝐸 𝐺 𝑢 𝑘 = 𝑑 2 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝑢 𝑘 , ∀ 1 ≤ 𝑘 ≤ 𝑛 . (3.21) Moreover, it can be easily verified that for every 𝑘 , ℓ ∈ { 1 , . . . , 𝑛 } , 𝑢 𝑇 𝑘 𝐺 𝑢 ℓ =  𝑑 𝑘 ( 𝐸 ⊕ 𝐺 )  𝑑 ℓ ( 𝐸 ⊕ 𝐺 ) 𝛿 𝑘 ℓ , (3.22) where 𝛿 𝑘 ℓ = 0 if 𝑘 ≠ ℓ , and 𝛿 𝑘 𝑘 = 1 . Define 𝑥 𝑘 : =  0 𝑢 𝑘  , 𝑦 𝑘 : =  − 1 𝑑 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝐺 𝑢 𝑘 0  ∀ 1 ≤ 𝑘 ≤ 𝑛 . (3.23) It can be verified using ( 3.22 ) and ( 3.23 ) that { 𝑥 1 , . . . , 𝑥 𝑛 , 𝑦 1 , . . . , 𝑦 𝑛 } forms a symplectic basis of R 2 𝑛 so that 𝑀 : = [ 𝑥 1 , . . . , 𝑥 𝑛 , 𝑦 1 , . . . , 𝑦 𝑛 ] ∈ SP 2 𝑛 . In fact, we have 𝑀 ∈ SP 2 𝑛 ( 𝐸 ⊕ 𝐺 ) . Indeed, by using the block forms of 𝑥 𝑘 , 𝑦 𝑘 given by ( 3.23 ), we have for each 𝑘 ∈ { 1 , . . . , 𝑛 } , ( 𝐸 ⊕ 𝐺 ) 𝑥 𝑘 = 𝑑 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝐽 𝑦 𝑘 , (3.24) ( 𝐸 ⊕ 𝐺 ) 𝑦 𝑘 = − 𝑑 𝑘 ( 𝐸 ⊕ 𝐺 ) 𝐽 𝑥 𝑘 . (3.25) Now, let 𝑘 ∈ { 1 , . . . , 𝑛 } be arbitrary and define 𝑊 : = [ 𝑥 1 , . . . 𝑥 𝑘 , 𝑦 1 , . . . , 𝑦 𝑘 ] . It is straightforward to verify that 𝑊 𝑇 𝐽 2 𝑛 𝑊 = 𝐽 2 𝑘 . Also, by the structure of 𝑊 and a routine computation with the trace operation, 2026/03/31 01:08 Weak supermajorization between symplectic spectra of positive denite matrix and its pinching 5 we get tr  𝑊 𝑇 ( 𝐸 ⊕ 𝐺 ) 𝑊  = tr  𝑊 𝑇 𝐴𝑊  . is gives 2 𝑘  𝑗 = 1 𝑑 𝑗 ( 𝐸 ⊕ 𝐺 ) = tr  𝑊 𝑇 ( 𝐸 ⊕ 𝐺 ) 𝑊  (3.26) = tr  𝑊 𝑇 𝐴𝑊  (3.27) ≥ min 𝑋 ∈ M 2 𝑛 , 2 𝑘 𝑋 𝑇 𝐽 2 𝑛 𝑋 = 𝐽 2 𝑘 tr  𝑋 𝑇 𝐴 𝑋  (3.28) = 2 𝑘  𝑗 = 1 𝑑 𝑗 ( 𝐴 ) . (3.29) e last equality is due to [ 12 , eorem 5]. is proves that 𝑑 ( 𝐸 ⊕ 𝐺 ) ≺ 𝑤 𝑑 ( 𝐴 ) . ■ For the symplectic pinching 𝒞 𝑠 relative to ( 1 , . . . , 1 ) ∈ R 𝑛 so that 𝒞 𝑠 ( 𝐸 ⊕ 𝐺 ) = Δ ( 𝐸 ) ⊕ Δ ( 𝐺 ) , we have 𝑑 ( 𝒞 𝑠 ( 𝐸 ⊕ 𝐺 ) ) = Δ 𝑠 ( 𝐴 ) by Lemma 3.1 , where Δ 𝑠 ( 𝐴 ) is defined in ( 2.8 ). It thus follows from eorem 9 of [ 12 ] that Δ 𝑠 ( 𝐴 ) ≺ 𝑤 𝑑 ( 𝐸 ⊕ 𝐺 ) . (3.30) e relations ( 3.13 ) and ( 3.30 ) together imply that Δ 𝑠 ( 𝐴 ) ≺ 𝑤 𝑑 ( 𝐴 ) . (3.31) e weak supermajorization relation in ( 3.31 ) is a symplectic analog of the classic Schur’s theorem established in [ 9 , eorem 1]. Furthermore, for any pinching 𝒞 , we appeal to eorem 9 of [ 12 ] again to write 𝑑 ( 𝒞 ( 𝐸 ) ⊕ 𝒞 ( 𝐺 ) ) ≺ 𝑤 𝑑 ( 𝐸 ⊕ 𝐺 ) . (3.32) Our result ( 3.13 ) together with ( 3.32 ) yields 𝑑 ( 𝒞 ( 𝐸 ) ⊕ 𝒞 ( 𝐺 ) ) ≺ 𝑤 𝑑 ( 𝐴 ) . (3.33) e following example is a striking illustration of the fact that the weak supermajorization relation in ( 3.13 ) may fail to hold for other types of pinching of 𝐴 . Example 3.3 Let 𝐴 =  7 6 6 7  ⊕  7 6 6 7  and consider the pinching 𝒞 ( 𝐴 ) =  7 0 0 7  ⊕  7 6 6 7  of 𝐴 relative to ( 1 , 3 ) . Computation yields 𝑑 ( 𝐴 ) = [ 1 , 13 ] 𝑇 and 𝑑 ( 𝒞 ( 𝐴 ) ) = [ 2 . 65 , 9 . 54 ] 𝑇 showing that 𝑑 ( 𝒞 ( 𝐴 ) ) is not weakly supermajorized by 𝑑 ( 𝐴 ) . We now turn to discussing a few consequences of our main result. Corollary 3.4 The eigenvalues of  𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2 are weakly supermajorized by the symplectic eigenvalues of 𝐴 , i.e., 𝜆   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  ≺ 𝑤 𝑑 ( 𝐴 ) . (3.34) Proof: By substituting ( 3.2 ) into ( 3.13 ), we get ( 3.34 ). ■ It is known [ 14 ] that 𝜆   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  ≤ 𝜆  𝐸 + 𝐺 2  . (3.35) 2026/03/31 01:08 Weak supermajorization between symplectic spectra of positive denite matrix and its pinching 6 e above inequality together with ( 3.34 ) implies that 𝜆  𝐸 + 𝐺 2  ≺ 𝑤 𝑑 ( 𝐴 ) . (3.36) Alternatively, we can also arrive at ( 3.36 ) by the following arguments. Observe that 𝑑 ( 𝐸 ⊕ 𝐺 ) = 𝑑 ( 𝐺 ⊕ 𝐸 ) . We have 𝜆 ( 𝐸 + 𝐺 ) = 𝑑 ( ( 𝐸 + 𝐺 ) ⊕ 𝑑 ( 𝐸 + 𝐺 ) ) (3.37) = 𝑑 ( ( 𝐸 ⊕ 𝐺 ) + ( 𝐺 ⊕ 𝐸 ) ) (3.38) ≺ 𝑤 𝑑 ( 𝐸 ⊕ 𝐺 ) + 𝑑 ( 𝐺 ⊕ 𝐸 ) (3.39) = 2 𝑑 ( 𝐸 ⊕ 𝐺 ) (3.40) ≺ 𝑤 2 𝑑 ( 𝐴 ) . (3.41) e equality ( 3.37 ) follows from ( 3.3 ), the weak supermajorization ( 3.39 ) is due to eorem 1 of [ 15 ], and ( 3.41 ) is from eorem 3.2 . Corollary 3.5 For any pinching 𝒞 , we have 𝜆   𝒞 ( 𝐺 ) 1 / 2 𝒞 ( 𝐸 ) 𝒞 ( 𝐺 ) 1 / 2  1 / 2  ≺ 𝑤 𝜆   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  . (3.42) In particular, 𝜆  𝒞 ( 𝐸 ) 1 / 2  ≺ 𝑤 𝜆  𝐸 1 / 2  . (3.43) Proof: By Lemma 3.1 , we have 𝑑 ( 𝐸 ⊕ 𝐺 ) = 𝜆   𝐺 1 / 2 𝐸 𝐺 1 / 2  1 / 2  , (3.44) and 𝑑 ( 𝒞 ( 𝐸 ) ⊕ 𝒞 ( 𝐺 ) ) = 𝜆   𝒞 ( 𝐺 ) 1 / 2 𝒞 ( 𝐸 ) 𝒞 ( 𝐺 ) 1 / 2  1 / 2  . (3.45) e weak supermajorization relation ( 3.42 ) then follows by substituting the above identities into ( 3.32 ). Also, ( 3.43 ) follows by choosing 𝐺 equal to the identity matrix in ( 3.42 ). ■ An alternative way to attain ( 3.43 ) is as follows. For any Hermitian matrix 𝐸 and any pinching 𝒞 , we have discussed that 𝜆 ( 𝒞 ( 𝐸 ) ) ≺ 𝜆 ( 𝐸 ) . (3.46) Noting that √ 𝑥 : [ 0 , ∞) − → [ 0 , ∞) is a concave function, an appeal to eorem 5.A.1 of [ 16 ] gives ( 3.43 ). 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