Let and be matrices. If , show that .
Find all matrices commuting with , where is the matrix all elements of which are equal to .
Prove the following statements.
(a) Let , where are distinct. If , then is a diagonal matrix.
(b) Let , where are distinct and nonzero, and be an matrix, where . If and , then .
證明 為正規矩陣 (normal matrix) 是一個充分條件。
Let and be normal matrices such that , where denotes the column space of . Prove that is a normal matrix. Note that is a normal matrix if .
任一正規矩陣 (normal matrix) 可表示為一個正規矩陣的平方。
Let be a normal matrix, i.e., . Prove that there exists a normal matrix such that .
證明 Hermitian 矩陣的秩與跡數不等式。
Let be an nonzero Hermitian matrix. Prove that
利用相合 (congruence) 變換證明若 ，則 。
Let and be Hermitian matrices. We will write that if is positive definite. The inequality means that is positive definite. Prove that if , then .
Let be a real orthogonal matrix and . Prove that