Category Archives: pow 線性方程與矩陣代數

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.

這是網友范智忠提供的問題。 Let and be matrices. If , show that .

找出所有的可交換矩陣。 Find all matrices commuting with , where is the matrix all elements of which are equal to .

若對角矩陣有相異對角元與某個矩陣是可交換的，則該矩陣也是對角矩陣。 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 .

證明一個特殊非負矩陣的逆矩陣也是非負矩陣。 Let be an real matrix. Show that and have all elements nonnegative if and only if each row and each column of has exactly one positive element and the rest of the elements are zeros.

這是關於半幻方 (semi-magic) 矩陣的分解式問題。 An matrix is said to be a semi-magic matrix if the sums of the rows and columns are all equal. Show that a semi-magic matrix can be decomposed as such that for integer , .

證明兩冪等矩陣 (idempotent matrix) 之差為冪等矩陣的一個充要條件。 Let and be idempotent matrices, i.e., and . Show that is idempotent if and only if .

利用冪等矩陣 (idempotent matrix) 計算分塊上三角矩陣的冪。 For the matrix , determine .

證明一個直覺的命題：若所有的 使得 ，則 。 Suppose that and are complex matrices. If holds for every , prove that .