1.2 Special Property Matrices (4)If Umxm is unitary,then UT,UH,U,U-1,Ui (i=1,2,...)are unitary. (5)U and V are unitary=UV is unitary. (6)If Umxm,Vnxn are unitary,then ①U⊕V unitary ②U⑧V unitary (7)If Umxm is unitary,then ①det(U0=±l: ②rank(U)=m: 3 U is normal,i.e.,UU=UHU: ④入is an eigenvalue of U→lN=l; ⑤xmx1→‖Ux2=‖l2; ⑥Amxn→‖UAlF=‖AlF ⑦Anxm→‖AUlF=‖A‖lF. 1 Special Matrices 7/601.2 Special Property Matrices (4) If Um×m is unitary, then UT, UH, U∗ , U−1 , Ui (i = 1, 2, · · ·) are unitary. (5) U and V are unitary ⇒ UV is unitary. (6) If Um×m, Vn×n are unitary, then 1 U ⊕ V unitary 2 U ⊗ V unitary (7) If Um×m is unitary, then 1 det(U) = ±1; 2 rank(U) = m; 3 U is normal, i.e., UUH = UHU; 4 λ is an eigenvalue of U ⇒ |λ| = 1; 5 xm×1 ⇒ kUxk2 = kxk2; 6 Am×n ⇒ kUAkF = kAkF; 7 An×m ⇒ kAUkF = kAkF. 1 Special Matrices 7 / 60