Tag Archives: Hermitian 矩陣

證明 Hermitian 矩陣的秩與跡數不等式。 Let be an nonzero Hermitian matrix. Prove that .

若一個 Hermitian 矩陣的主對角元為其特徵值，則此矩陣是對角矩陣。 Let be an Hermitian matrix whose eigenvalues, including multiple appearances, are the diagonal elements , . Prove that is diagonal.

這是關於相合變換 (congruence transformation) 的問題。 Let and be Hermitian matrices. Suppose is invertible. Show that there exists a nonsingular matrix so that and are diagonal if and only if is diagonalizable and all its eigenvalues are real.

這是 Hermitian 矩陣的一個充分條件證明問題 (事實上，這是一個充要條件)。 Let be an complex matrix. Show that if is real for every , then .

這是實對稱矩陣的二次型極值問題。 Let be an Hermitian matrix. Let and be the smallest and largest eigenvalues of , respectively. For every unit vector , show that .

本週問題是證明 Hermitian 正定矩陣的積必可對角化。 Let and be Hermitian matrices. Show that if or is positive definite, then is diagonalizable.

這是證明半正定矩陣和 Hermitian 矩陣乘積的特徵值必為實數。 Let and be Hermitian matrices. Prove that following statements. (a) If or is positive semidefinite, then all the eigenvalues of are real. (b) If and are positive semidefinite, then all the eigenvalues of are nonnegative.