Tag Archives: 半正定矩陣

這是兩個實對稱矩陣以相合變換同時可對角化問題。 Let and be real symmetric matrices, and , . If there exists a such that is a positive semidefinite matrix and , then there exists a nonsingular matrix such that both and are diagonal. Note that denotes the nullspace … Continue reading →

證明兩個半正定矩陣之和的行列式大於或等於兩矩陣的行列式之和。 Let and be Hermitian and positive semidefinite matrices. Show that .

若半正定矩陣的一個主對角元等於零，則該列與行的所有元必為零。 Let be an Hermitian matrix. If is positive semidefinite and for some , show that for all .

這是關於半正定矩陣的二次型等於零的問題。 Let be a real symmetric positive semidefinite matrix. If , show that .

實對稱矩陣與反對稱矩陣的平方有甚麼性質？ Let be an real matrix. Prove the following statements. (a) If is symmetric, then is positive semidefinite. (b) If is skew-symmetric, then is positive semidefinite.