Author Archives: ccjou

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 .

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 .

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 .

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 .

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.

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.