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At first glance, Corollary 1.18 (and hence Theorem 1.16) is strange: For positive numbers a ≥ b, we have a2 ≥ (ba2 b)1/2 ≥ b2 . We know the matrix analog that A ≥ B ≥ 0 implies A2 ≥ B 2 is false, but Corollary 1.18 asserts that the matrix analog of the stronger inequality (ba2 b)1/2 ≥ b2 holds. This example shows that when we move from the commutative world to the noncommutative one, direct generalizations may be false, but a judicious modification may be true. Notes and References. Corollary 1.18 is a conjecture of N. N. Chan and M. K. Kwong [29]. T. Furuta [38] solved this conjecture by proving the more general Theorem 1.16. See [39] for a related result....
Then Ψ is again a doubly stochastic map and diag(y1 , . . . , yn ) = Ψ (diag(x1 , . . . , xn )). (2.11)...
Applying Lemma 3.18 again with G = T ∗ T /2, H = T T ∗ /2 gives {(s2 + s2 −j +1 )/2} ≺ {sj (A2 + B 2 )} ≺ {s2 }. j n j Combining (3.30) and (3.31) we get (3.28) and (3.29). A n important class of unitarily invariant norms are Schatten p-norms. They are the lp norms of the singular values: ⎛ A
p...
Since weak log-ma jorization implies ma jorization (Theorem 2.7), we get {|λj |p } ≺w {sp } j In particular, jn
=1...
(4.13)
Theorem 4.10 Let f (t) be a nonnegative operator monotone function on [0, ∞) and · be a normalized unitarily invariant norm. Then for every matrix A,
4.1 Operator Monotone Functions
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f ( A ) ≤ f (|A|) ...
4.4 Inequalities of H¨lder and Minkowski Types o
83
Pro of. In Theorem 4.30, replacing A, B , X and t by A2 , B −2 , A−1 X B and (1 + s)/2 respectively, we see that ψ (s) is convex on (−1, 1), decreasing on (−1, 0), increasing on (0, 1) and attains its minimum at s = 0 when −1 ≤ s ≤ 1. Next replacing A, B by their appropriate powers it is easily seen that the above convexity and monotonicity of ψ (s) on those intervals are equivalent to tGe same properties on (−∞, ∞), (−∞, 0) and (0, ∞) respectively. h iven a norm A is defined as · on Mn , the condition number of an invertible matrix c(A) = A · A−1 ...
(4.63)
Pro of. Denote by Nui the set of unitarily invariant norms on Mn . By Fan’s dominance principle (Lemma 4.2) we have max{ Φ : · ∈ Nui } = max{ Φ
(k )
: 1 ≤ k ≤ n}...
This proves (4.69). The examples with Ψ , I , B in the proof of Theorem 4.38 also show both (4.6T) and (4.69) are sharp. 8 he idea in the above proofs is simple: The Frobenius norm is invariant under permutations of matrix entries. The next result shows that the converse is also true...
102
5. The van der Waerden Conjecture
Pro of. By Lemma 5.2, perA(i|j ) = 0 for some i, j if and only if the submatrix A(i|j ) and hence A contains an s × t zero submatrix with s + t = n. In other words, perA(i|j ) = 0 for some pair i, j if and only if A is partly decomposable. L emma 5.5 If A is a partly decomposable doubly stochastic matrix, then there exist permutation matrices P, Q such that P AQ is a direct sum of doubly stochastic matrices. Pro of. Since A is partly decomposable, there exist permutation matrices P an w d Q such that B C P AQ = 0D here B and D are square. Using the same argument as in the proof of CorDllary 5.3 it is easy to show C = 0. o enote by e ∈ Rn the vector with all components equal to 1. Then the condition that all the row and column sums of an n-square matrix A be 1 can be expressed as Ae = e and AT e = e. From this it is obvious that if A is doubly stochastic then so are AT A and AAT . Lemma 5.6 If A is a ful ly indecomposable doubly stochastic matrix, then so are AT A and AAT . Pro of. We have already remarked that both AT A and AAT are doubly stochastic. Suppose AT A is partly decomposable and A is n × n, i.e., there exist nonempty α, β ⊂ N such that |α| + |β | = n and AT A[α|β ] = 0. The latter condition is equivalent to (A[N |α])T A[N |β ] = 0, which implies that the sum of the numbers of zero rows of A[N |α] and of A[N |β ] is at least n. Let t be the number of zero rows of A[N |α]. If t ≥ n − |α|, then A[N |α] and hence A has an |α| × (n − |α|) zero submatrix, which contradicts the assumption that A is fully indecomposable. But if t < n − |α| (= |β |), the number s of zero rows of A[N |β ] satisfies s ≥ n − t > n − |β |, so that A is partly decomposable, which again contradicts the assumption. L In the same way we can show that AAT is fully indecomposable. emma 5.7 If A is a ful ly indecomposable doubly stochastic matrix and Ax = x for some real vector x, then x is a constant multiple of e...
be any positive number smaller than min(bkk , c11 ) and define ˜ G( ) ≡ A − (Ekk + Ek+1,k+1 ) + (Ek,k+1 + Ek+1,k )...
By Lemma 5.8, A is fully indecomposable. Therefore both AT A and AAT are fully indecomposable and doubly stochastic by Lemma 5.6. Then applying Lemma 5.7 to (5.6) we deduce that both λ and μ are constant multiples of e, say, λ = ce and μ = de. It follows from (5.5) that...
each an−2 (i) > 0 and (5.14) we have per(a1 , . . . , an−3 , c, c, ei ) = 0 1 ≤ i ≤ n. (5.15)...
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