每週問題 November 3, 2014

實對稱矩陣與反對稱矩陣的平方有甚麼性質?

Let A be an n\times n real matrix. Prove the following statements.

(a) If A is symmetric, then A^2 is positive semidefinite.
(b) If A is skew-symmetric, then -A^2 is positive semidefinite.

 
參考解答:

(a) 若 A^T=A,對於任意 \mathbf{x}\in\mathbb{R}^n

\displaystyle  \mathbf{x}^TA^2\mathbf{x}=\mathbf{x}^T(A^TA)\mathbf{x}=(A\mathbf{x})^T(A\mathbf{x})=\Vert A\mathbf{x}\Vert^2\ge 0

證明 A^2 是半正定。

(b) 若 A^T=-A,對於任意 \mathbf{x}\in\mathbb{R}^n

\displaystyle  \mathbf{x}^T(-A^2)\mathbf{x}=\mathbf{x}^T(A^TA)\mathbf{x}=(A\mathbf{x})^T(A\mathbf{x})=\Vert A\mathbf{x}\Vert^2\ge 0

證明 -A^2 是半正定。

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