B***y
发帖数: 83 | 1 这是道非常有意思的题目。涉及到凸集结构。
首先在一个 Banach space B 中，
我们定义一个凸集 C 为
1) If x, y \in C, then for t \in [0, 1], tx + (1-t)y \in C.
现在定义 C = set of all n*n seme-positive definite matrix A = (aij), with each
element aij \in [0, 1].
定义 C_k = set of all n*n seme-positive definite matrix A = (aij), with each
element aij \in [0, 1], with rank >= k.
那么 C 在所有 n*n 对称矩阵中，给出了一个凸集。更重要的是 C_k 也是凸集。
凸集的一般事实是：如果存在于有限维空间中，则总存在 basis, 即线性
无关向量集。( reference: 张恭庆，关肇直：线性泛函分析讲义，convex analysis,
1977
J.T. Marti.)
Theorem ( Minkows | o**p
发帖数: 2 | 2 I think what Boll means is: You got k(k+1)/2 rank 1 matrix
decomposition of the original matrix. And each rank 1 matrix can
be represented by the vector.
For your answer. Yes, it's easy to get the 'spectral' decomposition.
But those factors may not be satisfy the requirment [0,1], as you
mentioned. But this requirment is important.
each
each
analysis,
\in C,
+ S
norm, |
|
