每週問題 June 26, 2017

對於秩-1方陣 ，證明 。

Let be an matrix and . Prove that .

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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.

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 .

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.

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 .

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

.

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 .

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 .

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.

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.