By R.A. Kalnin

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34. 35 Let Z, W be n-square nonnegative matrices. Then (i) Z is doubly superstochastic if and only if Z(α, β) ≥ |α| + |β| − n for all α, β ⊆ N ; (ii) W is doubly substochastic if and only if all row sums and column sums of W are less than or equal to 1. For A = (aij ) ∈ Mn and a real number p > 0, we denote A|◦|p ≡ (|aij |p ) ∈ Mn . The following entrywise inequalities involve the smallest and the largest singular values. 36 Let A ∈ Mn and let p, q be real numbers with 0 < p ≤ 2 and q ≥ 2. 5 Singular Values and Matrix Entries A|◦|q ≤e s1 (A)q C.

Then sj (A − B) ≤ sj (A ⊕ B), j = 1, 2, . . , n. 23) First Proof. Note that s(A ⊕ B) = s(A) ∪ s(B). It is easily veriﬁed (say, by using the spectral decompositions of A, B) that for a ﬁxed j with 1 ≤ j ≤ n there exist H, F ∈ Mn satisfying 0 ≤ H ≤ A, 0 ≤ F ≤ B, rankH + rankF ≤ j − 1 and sj (A ⊕ B) = (A − H) ⊕ (B − F ) ∞ . Thus sj (A ⊕ B) = max{ A − H denote by I the identity matrix. Since ∞, B−F ∞} ≡ γ. 22) we have sj (A − B) ≤ A − B − (H − F ) ∞ γ γ = (A − H − I) − (B − F − I) ∞ 2 2 γ γ ≤ (A − H) − I ∞ + (B − F ) − I 2 2 γ γ ≤ + = γ = sj (A ⊕ B).

4 Matrix Cartesian Decompositions λj (U ∗ AU ) ≥ λj+k (A), and 45 1 ≤ j ≤ n − k, λj (B) ≥ λj (U ∗ BU ) ≥ λj+k (B), 1 ≤ j ≤ n − k. The ﬁrst of these inequalities shows that n−k ∗ n ∗ [λj (U AU )] ≥ 2 tr (U AU ) = 2 j=1 αj2 . 24 to the second inequality, we deduce that there exist n − 2k distinct indices j1 , . . , jn−2k such that |λjs (U ∗ BU )| ≥ |λk+s (B)|, Therefore s = 1, . . , n − 2k. n−k ∗ n ∗ [λj (U BU )] ≥ 2 tr (U BU ) = 2 j=1 βj2 . 45). Now let n/2 ≤ k ≤ n. 42) is equivalent to n n s2j ≥ j=k+1 αj2 .