v2010.10.26 - Convex Optimization

D.2. TABLES OF GRADIENTS AND DERIVATIVES 693D.2.3trace∇ x µ x = µI ∇ X tr µX = ∇ X µ trX = µI∇ x 1 T δ(x) −1 1 = ddx x−1 = −x −2∇ x 1 T δ(x) −1 y = −δ(x) −2 y∇ X tr X −1 = −X −2T∇ X tr(X −1 Y ) = ∇ X tr(Y X −1 ) = −X −T Y T X −Tddx xµ = µx µ−1 ∇ X tr X µ = µX µ−1 , X ∈ S M∇ X tr X j = jX (j−1)T∇ x (b − a T x) −1 = (b − a T x) −2 a∇ X tr ( (B − AX) −1) = ( (B − AX) −2 A ) T∇ x (b − a T x) µ = −µ(b − a T x) µ−1 a∇ x x T y = ∇ x y T x = y∇ X tr(X T Y ) = ∇ X tr(Y X T ) = ∇ X tr(Y T X) = ∇ X tr(XY T ) = Y∇ X tr(AXBX T ) = ∇ X tr(XBX T A) = A T XB T + AXB∇ X tr(AXBX) = ∇ X tr(XBXA) = A T X T B T + B T X T A T∇ X tr(AXAXAX) = ∇ X tr(XAXAXA) = 3(AXAXA) T∇ X tr(Y X k ) = ∇ X tr(X k Y ) = k−1 ∑ (X i Y X k−1−i) Ti=0∇ X tr(Y T XX T Y ) = ∇ X tr(X T Y Y T X) = 2 Y Y T X∇ X tr(Y T X T XY ) = ∇ X tr(XY Y T X T ) = 2XY Y T∇ X tr ( (X + Y ) T (X + Y ) ) = 2(X + Y ) = ∇ X ‖X + Y ‖ 2 F∇ X tr((X + Y )(X + Y )) = 2(X + Y ) T∇ X tr(A T XB) = ∇ X tr(X T AB T ) = AB T∇ X tr(A T X −1 B) = ∇ X tr(X −T AB T ) = −X −T AB T X −T∇ X a T Xb = ∇ X tr(ba T X) = ∇ X tr(Xba T ) = ab T∇ X b T X T a = ∇ X tr(X T ab T ) = ∇ X tr(ab T X T ) = ab T∇ X a T X −1 b = ∇ X tr(X −T ab T ) = −X −T ab T X −T∇ X a T X µ b = ...