Loewner order

\( \renewcommand{\vec}[1]{\boldsymbol{#1}} \DeclareMathOperator*{\E}{\mathbb{E}} \DeclareMathOperator*{\Var}{\mathrm{Var}} \DeclareMathOperator*{\Cov}{\mathrm{Cov}} \DeclareMathOperator*{\argmin}{\mathrm{arg\,min\;}} \DeclareMathOperator*{\argmax}{\mathrm{arg\,max\;}} \def\ZZ{{\mathbb Z}} \def\NN{{\mathbb N}} \def\RR{{\mathbb R}} \def\CC{{\mathbb C}} \def\QQ{{\mathbb Q}} \def\FF{{\mathbb FF}} \def\EE{{\mathbb E}} \newcommand{\tr}{{\rm tr}} \newcommand{\sign}{{\rm sign}} \newcommand{\1}{𝟙} \newcommand{\inprod}[2]{\left\langle #1, #2 \right\rangle} \newcommand{\set}[1]{\left\{#1\right\}} \require{physics} \)

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: \[\lambda_k(A) = \max_{\substack{\text{subspace }S\subseteq \mathbb{R}^n\\ \text{dim} S=k}} \min_{\substack{x\in S\\x\neq 0}} \frac{x^T Ax}{x^T x} \geq \max_{\substack{\text{subspace }S\subseteq \mathbb{R}^n\\ \text{dim} S=k}} \min_{\substack{x\in S\\x\neq 0}} \frac{x^T Bx}{x^T x} = \lambda_k(B)\]


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