CONCAVITY OF CERTAIN MAPS ON POSITIVE DEFINITE MATRICES AND APPLICATIONS TO HADAMARD PRODUCTS
CONCAVITY OF CERTAIN MAPS ON POSITIVE DEFINITE MATRICES AND APPLICATIONS TO HADAMARD PRODUCTS
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DOI：
10.1016/0024-3795(79)90179-4
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发表时间：
1979-01-01
影响因子：
1.1
通讯作者：
ANDO, T
中科院分区：
文献类型：
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作者：
ANDO, T
如果 f 是 (0,∞) 上的正函数，且在 Löwner 意义上对于每个 n 都是 n 阶单调，并且如果 Φ 1 和 Φ 2 是正定矩阵之间的凹映射，则以下涉及张量积的映射：(A, B)↦ f [Φ 1 (A)− 1⊗ Φ 2 (B)]·(Φ 1 (A)⊗ I) 被证明是凹的。如果 Φ 1 是仿射的，则无需使用正性就可以证明映射 (A, B)↦ f [Φ 1 (A)⊗ Φ 2 (B)− 1]·(Φ 1 (A)⊗ I) 是凸的。这些产生映射 (A, B)↦ A 1− p⊗ B p (0< p⩽ 1)（利布定理）的凹性和映射 (A, B)↦ A 1+ p⊗ B− p (0< p⩽ 1) 的凸性，以及映射 (A, B)↦(A· log [A])⊗ I− 的凸性A⊗ 日志 [B]。然后应用这些凹凸定理从上到下获得正定矩阵的 Hadamard 乘积的异常估计。
If f is a positive function on (0,∞) which is monotone of order n for every n in the sense of Löwner and if Φ 1 and Φ 2 are concave maps among positive definite matrices, then the following map involving tensor products:(A, B)↦ f [Φ 1 (A)− 1⊗ Φ 2 (B)]·(Φ 1 (A)⊗ I) is proved to be concave. If Φ 1 is affine, it is proved without use of positivity that the map (A, B)↦ f [Φ 1 (A)⊗ Φ 2 (B)− 1]·(Φ 1 (A)⊗ I) is convex. These yield the concavity of the map (A, B)↦ A 1− p⊗ B p (0< p⩽ 1)(Lieb's theorem) and the convexity of the map (A, B)↦ A 1+ p⊗ B− p (0< p⩽ 1), as well as the convexity of the map (A, B)↦(A· log [A])⊗ I− A⊗ log [B]. These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices.