Category Archives: pow 內積空間

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 .

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

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 .

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.

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 .