每週問題 January 23, 2017

證明正定矩陣的伴隨矩陣 (adjugate) 也是一個正定矩陣。

Prove that if A is a real symmetric positive definite then \hbox{adj}A is also a symmetric positive definite matrix.

 
參考解答:

假設 A 是一個 n\times n 階實對稱正定矩陣。寫出正交對角化表達式 A=QDQ^T,其中 Q 是一個實正交矩陣,Q^T=Q^{-1}D=\hbox{diag}(\lambda_1,\ldots,\lambda_n)\lambda_i>0。使用伴隨矩陣性質,

\begin{aligned} \hbox{adj}A&=\hbox{adj}(QDQ^T)=(\hbox{adj}Q^T)(\hbox{adj}D)(\hbox{adj}Q)\\ &=(\det Q^T)(Q^T)^{-1}(\hbox{adj}D)(\det Q)Q^{-1}\\ &=(\det Q)^{-1}Q(\hbox{adj}D)(\det Q)Q^T\\ &=Q(\hbox{adj}D)Q^T, \end{aligned}

其中

\hbox{adj}D=\begin{bmatrix} \prod_{i\neq 1}\lambda_i&&\\ &\ddots&\\ &&\prod_{i\neq n}\lambda_i \end{bmatrix}

是一個對角矩陣且主對角元皆為正數,因此得證。

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