對於秩-1方陣 ，證明 。
Let be an matrix and . Prove that .
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.
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 .
Let and be matrices, where is an odd number. Prove that if then at least one of the matrices and is singular.
Let and be complex matrices of size and , respectively. If , prove that for some matrix .
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 .
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.