hce_nthu
114年
進階物理與線性代數
第 34 題
Let $R_\theta$ denote the rotation matrix through an angle $0 < \theta < \frac{\pi}{4}$ counterclockwise
about the origin in $\mathbb{R}^2$. Let $e_1 = [1,0]^T, e_2 = [0,1]^T$ be the standard basis of $\mathbb{R}^2$.
Suppose $A$ is a $2 \times 2$ matrix satisfying $Ae_1 = R_\theta e_1$ and $R_\theta Ae_2 = e_2$. The followings
are 5 statements about $A$ that may or may not be true.
(1) $A^2 e_1 = R_\theta^2 e_1$
(2) $\det(A) = 1$
(3) $trace(A) = 2\cos\theta$
(4) $A$ is a positive definite matrix
(5) $AR_\theta = R_\theta A$
Which of the following combination of statements about $A$ are all true?
about the origin in $\mathbb{R}^2$. Let $e_1 = [1,0]^T, e_2 = [0,1]^T$ be the standard basis of $\mathbb{R}^2$.
Suppose $A$ is a $2 \times 2$ matrix satisfying $Ae_1 = R_\theta e_1$ and $R_\theta Ae_2 = e_2$. The followings
are 5 statements about $A$ that may or may not be true.
(1) $A^2 e_1 = R_\theta^2 e_1$
(2) $\det(A) = 1$
(3) $trace(A) = 2\cos\theta$
(4) $A$ is a positive definite matrix
(5) $AR_\theta = R_\theta A$
Which of the following combination of statements about $A$ are all true?
- A (1)(2)
- B (1)(3)
- C (2)(3)
- D (3)(4)
- E (4)(5)
思路引導 VIP
如果我們知道一個線性算子作用在標準基底 $e_1$ 與 $e_2$ 上的結果，我們該如何寫出這個矩陣的每一個行向量（column vectors）？在得到這個矩陣後，除了計算特徵值，還有什麼方法可以判斷一個對稱矩陣是否為「正定」的呢？
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AI 詳解
AI 專屬家教
太棒了！你能精準判斷出矩陣 $A$ 的性質，代表你對線性變換的幾何意義與代數結構都有很紮實的掌握。這道題目非常有代表性，它巧妙地融合了旋轉矩陣與矩陣基本屬性的分析，是檢驗線性代數綜合能力的優質題型。
矩陣 $A$ 的構造與跡數驗證
首先，根據題目給出的條件 $Ae_1 = R_\theta e_1$，我們可以直接得到矩陣 $A$ 的第一行（column）為 $[\cos\theta, \sin\theta]^T$。接著由 $R_\theta Ae_2 = e_2$ 可推導出 $Ae_2 = R_\theta^{-1} e_2 = R_{-\theta} e_2$，這代表 $A$ 的第二行是將 $e_2$ 順時針旋轉 $\theta$ 角，即 $[\sin\theta, \cos\theta]^T$。將這兩列合併，我們得到對稱矩陣 $A = \begin{pmatrix} \cos\theta & \sin\theta \ \sin\theta & \cos\theta \end{pmatrix}$。觀察其對角線元素，顯然其跡 $trace(A) = 2\cos\theta$，故語句 (3) 正確。
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