Author Archives: ccjou

每週問題 June 26, 2017

對於秩-1方陣 ,證明 。 Let be an matrix and . Prove that . Advertisements

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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.

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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 June 5, 2017

證明 是不可逆矩陣的一個充分條件。 Let and be matrices, where is an odd number. Prove that if then at least one of the matrices and is singular.

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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 May 22, 2017

以伴隨矩陣的行列式表達分塊矩陣的行列式。 Suppose is , is , is , and is a number. Prove that .

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每週問題 May 15, 2017

反對稱矩陣的伴隨矩陣 (adjugate) 是對稱或反對稱矩陣。 Let be an skew-symmetric matrix. Prove that is a symmetric matrix for odd and a skew-symmetric matrix for even .

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每週問題 May 8, 2017

這是關於基底的一個充分條件問題。 Let be vectors in () such that for . Prove that any of these vectors form a basis of .

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每週問題 May 1, 2017

證明嚴格對角佔優 (strictly diagonally dominant) 矩陣是可逆矩陣。 Let be an matrix. Prove that if for , then is invertible.

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每週問題 April 24, 2017

證明矩陣積的值域與零空間的維數恆等式。 Let be an matrix and be an matrix. Prove that . Note that and denote the column space and nullspace of , respectively.

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