Category Archives: pow 內積空間

每週問題 November 23, 2015

證明實正交投影矩陣的主對角元性質。 Let be an real orthogonal projection matrix, i.e., . Show that for all , and .

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每週問題 September 21, 2015

若 是正交矩陣,則 是反對稱矩陣。 Let be an orthogonal matrix, where each entry is a differentiable function of . Show that is skew-symmetric.

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每週問題 June 22, 2015

本週問題是證明正交變換的一些性質。 Let be a linear operator on an inner product space of dimension . If is an orthogonal transformation, i.e., , for every , prove the following statements. (a) for every . (b) for every eigenvalue of . (c) If … Continue reading

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每週問題 April 20, 2015

證明 Cayley 變換將正交矩陣映至反對稱矩陣。 Let be a real orthogonal matrix and . Show that Cayley transformation is skew-symmetric.

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每週問題 April 6, 2015

證明正交變換是一個線性變換。 If is a mapping on an inner product satisfying for all , show that is a linear transformation. Such a is called an orthogonal transformation.

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每週問題 March 2, 2015

這是關於實正交矩陣的特徵向量性質。 Let be a real orthogonal matrix, i.e., is real and . Let be an eigenvalue of and be a corresponding eigenvector, where and are real vectors. If is not real, show that and .

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每週問題 February 23, 2015

利用內積空間的基底表達兩個向量的內積。 If is an orthonormal basis for an inner-product space , show that for every .

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每週問題 February 2, 2015

本週問題關於 Cayley 變換。 Let be a skew Hermitian matrix. Show that Cayley transformation is a unitary matrix.

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每週問題 January 26, 2015

證明 階對合矩陣 (involutory matrix) 與冪等矩陣 (idempotent matrix) 具有一對一的關係。 A matrix satisfying is said to be an involutory matrix, and a matrix satisfying is said to be an idempotent matrix. Show that there is a one-to-one correspondence between the set of … Continue reading

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每週問題 December 29, 2014

這是最小平方法應用於矩陣之線性變換的問題。 Let be the vector space formed by real matrices. Let and . Consider the following transformation from to : . Determine a matrix that minimizes , where denotes Frobenius norm.

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