Category Archives: pow 內積空間

每週問題 June 19, 2017

證明一個可逆矩陣存在 QR 分解。 Prove that an invertible matrix can be represented in the form , where is an orthogonal matrix and is an upper triangular matrix. Advertisements

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每週問題 June 12, 2017

證明 Gram-Schmidt 正交化定理。 Let be a basis of an inner product space. Prove that there exists an orthogonal basis such that for all .

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每週問題 May 29, 2017

這是零空間的包容關係與矩陣乘法的問題。 Let and be complex matrices of size and , respectively. If , prove that for some matrix .

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每週問題 February 13, 2017

證明遍歷定理 (ergodic theorem)。 Let be a unitary matrix, i.e., . Prove that , where is the Hermitian projection matrix onto .

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每週問題 November 14, 2016

證明一個常見於多變量統計學的矩陣 是半正定。 Let and let be a complex matrix of rank . Show that the Hermitian matrix is positive semidefinite.

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每週問題 September 5, 2016

推導 Tikhonov 正則化 (regularization) 的最佳解。 Let be an real matrix and . Solve the Tikhonov regularization problem: , where .

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每週問題 June 6, 2016

若線性方程 是一致的,則 的行空間 (column space) 存在唯一一個解。 Let be an complex matrix. If is consistent for some , prove that there exists a unique solution in the column space of .

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每週問題 May 2, 2016

證明正規方程 (normal equation) 是一致的,意指存在解。 Let be an complex matrix. Show that for any , the normal equation is consistent, meaning that it has solutions.

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每週問題 April 11, 2016

利用畢氏定理判定兩個正交的複向量。 Let be a complex inner product space. Show that two vectors and in are orthogonal if and only if for all pairs of scalars and .

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每週問題 March 7, 2016

Schwarz 不等式的等號成立的一個充要條件為兩個向量是線性相關的。 Let and be vectors in an inner product space, and denote the inner product of and . Prove that if (that is, the Schwarz inequality reduces to an equality), then and are linearly dependent.

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