Written by
Li Zhuohua
on
Loewner order
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对称矩阵\(A\),\(B\)，\(A \succcurlyeq B \Leftrightarrow A-B \succcurlyeq 0\)，也等价于\(\forall v, v^T A v \geq v^T B v\)
定理：若\(A \succcurlyeq B\)，则\(A,B\)的特征值满足（假设特征值已排序）：
\[\lambda_i(A)\geq \lambda_i(B)\]
证明：
根据Courant-Fischer Theorem定理：
\[\lambda_k(A) = \max_{\substack{\text{subspace }S\subseteq \mathbb{R}^n\\ \text{dim} S=k}} \min_{\substack{x\in S\\x\neq 0}} \frac{x^T Ax}{x^T x} \geq \max_{\substack{\text{subspace }S\subseteq \mathbb{R}^n\\ \text{dim} S=k}} \min_{\substack{x\in S\\x\neq 0}} \frac{x^T Bx}{x^T x} = \lambda_k(B)\]