v2010.10.26 - Convex Optimization

692 APPENDIX D. MATRIX CALCULUSalgebraic continuedd(X+ t Y ) = Ydtddt BT (X+ t Y ) −1 A = −B T (X+ t Y ) −1 Y (X+ t Y ) −1 Addt BT (X+ t Y ) −T A = −B T (X+ t Y ) −T Y T (X+ t Y ) −T Addt BT (X+ t Y ) µ A = ..., −1 ≤ µ ≤ 1, X, Y ∈ S M +B T (X+ t Y ) −1 A = 2B T (X+ t Y ) −1 Y (X+ t Y ) −1 Y (X+ t Y ) −1 Adt 2 d 3B T (X+ t Y ) −1 A = −6B T (X+ t Y ) −1 Y (X+ t Y ) −1 Y (X+ t Y ) −1 Y (X+ t Y ) −1 Adt 3d 2ddt((X+ t Y ) T A(X+ t Y ) ) = Y T AX + X T AY + 2tY T AY(d 2dt (X+ t Y ) T A(X+ t Y ) ) = 2Y T AY2((X+ t Y ) T A(X+ t Y ) ) −1ddtddt= − ( (X+ t Y ) T A(X+ t Y ) ) −1(Y T AX + X T AY + 2tY T AY ) ( (X+ t Y ) T A(X+ t Y ) ) −1((X+ t Y )A(X+ t Y )) = YAX + XAY + 2tYAYd 2dt 2 ((X+ t Y )A(X+ t Y )) = 2YAYD.2.1.0.1 Exercise. Expand these tables.Provide unfinished table entries indicated by ... throughoutD.2.D.2.1.0.2 Exercise. log. (D.1.7)Find the first four terms of the Taylor series expansion for log x about x = 1.Prove that log x ≤ x −1; alternatively, plot the supporting hyperplane tothe hypograph of log x at[ xlog x]=[ 10]. D.2.2trace Kronecker∇ vec X tr(AXBX T ) = ∇ vec X vec(X) T (B T ⊗ A) vec X = (B ⊗A T + B T ⊗A) vec X∇ 2 vec X tr(AXBXT ) = ∇ 2 vec X vec(X)T (B T ⊗ A) vec X = B ⊗A T + B T ⊗A