v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
696 APPENDIX D. MATRIX CALCULUSD.2.5determinant∇ X detX = ∇ X detX T = det(X)X −T∇ X detX −1 = − det(X −1 )X −T = − det(X) −1 X −T∇ X det µ X = µ det µ (X)X −T∇ X detX µ = µ det(X µ )X −T∇ X detX k = k det k−1 (X) ( tr(X)I − X T) , X ∈ R 2×2∇ X detX k = ∇ X det k X = k det(X k )X −T = k det k (X)X −T∇ X det µ (X+ t Y ) = µ det µ (X+ t Y )(X+ t Y ) −T∇ X det(X+ t Y ) k = ∇ X det k (X+ t Y ) = k det k (X+ t Y )(X+ t Y ) −Tddet(X+ t Y ) = det(X+ t Y ) tr((X+ t Y dt )−1 Y )d 2dt 2 det(X+ t Y ) = det(X+ t Y )(tr 2 ((X+ t Y ) −1 Y ) − tr((X+ t Y ) −1 Y (X+ t Y ) −1 Y ))ddet(X+ t Y dt )−1 = − det(X+ t Y ) −1 tr((X+ t Y ) −1 Y )d 2dt 2 det(X+ t Y ) −1 = det(X+ t Y ) −1 (tr 2 ((X+ t Y ) −1 Y ) + tr((X+ t Y ) −1 Y (X+ t Y ) −1 Y ))ddt detµ (X+ t Y ) = µ det µ (X+ t Y ) tr((X+ t Y ) −1 Y )D.2.6logarithmicMatrix logarithm.dlog(X+ t Y dt )µ = µY (X+ t Y ) −1 = µ(X+ t Y ) −1 Y ,dlog(I − t Y dt )µ = −µY (I − t Y ) −1 = −µ(I − t Y ) −1 YXY = Y X[203, p.493]
D.2. TABLES OF GRADIENTS AND DERIVATIVES 697D.2.7exponentialMatrix exponential. [77,3.6,4.5] [331,5.4]∇ X e tr(Y T X) = ∇ X dete Y TX = e tr(Y T X) Y (∀X,Y )∇ X tre Y X = e Y T X T Y T = Y T e XT Y T∇ x 1 T e Ax = A T e Ax∇ x 1 T e |Ax| = A T δ(sgn(Ax))e |Ax| (Ax) i ≠ 0∇ x log(1 T e x ) = 11 T e x ex∇ 2 x log(1 T e x ) = 11 T e x (δ(e x ) − 11 T e x ex e xT )k∏∇ x x 1 ki = 1 (∏ kx 1 ki=1 k i)1/xi=1∇ 2 xk∏x 1 ki = − 1 ( k)(∏x 1 ki=1 k i δ(x) −2 − 1 )i=1 k (1/x)(1/x)Tddt etY = e tY Y = Y e tYddt eX+ t Y = e X+ t Y Y = Y e X+ t Y ,d 2dt 2 e X+ t Y = e X+ t Y Y 2 = Y e X+ t Y Y = Y 2 e X+ t Y ,XY = Y XXY = Y Xd jdt j e tr(X+ t Y ) = e tr(X+ t Y ) tr j (Y )
- Page 645 and 646: B.4. AUXILIARY V -MATRICES 64514. [
- Page 647 and 648: B.5. ORTHOGONAL MATRIX 647Given X
- Page 649 and 650: B.5. ORTHOGONAL MATRIX 649Figure 15
- Page 651 and 652: B.5. ORTHOGONAL MATRIX 651which is
- Page 653 and 654: Appendix CSome analytical optimal r
- Page 655 and 656: C.2. TRACE, SINGULAR AND EIGEN VALU
- Page 657 and 658: C.2. TRACE, SINGULAR AND EIGEN VALU
- Page 659 and 660: C.2. TRACE, SINGULAR AND EIGEN VALU
- Page 661 and 662: C.3. ORTHOGONAL PROCRUSTES PROBLEM
- Page 663 and 664: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 665 and 666: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 667 and 668: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 669 and 670: Appendix DMatrix calculusFrom too m
- Page 671 and 672: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 673 and 674: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 675 and 676: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 677 and 678: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 679 and 680: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 681 and 682: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 683 and 684: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 685 and 686: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 687 and 688: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 689 and 690: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 691 and 692: D.2. TABLES OF GRADIENTS AND DERIVA
- Page 693 and 694: D.2. TABLES OF GRADIENTS AND DERIVA
- Page 695: D.2. TABLES OF GRADIENTS AND DERIVA
- Page 699 and 700: Appendix EProjectionFor any A∈ R
- Page 701 and 702: 701U T = U † for orthonormal (inc
- Page 703 and 704: E.1. IDEMPOTENT MATRICES 703where A
- Page 705 and 706: E.1. IDEMPOTENT MATRICES 705order,
- Page 707 and 708: E.1. IDEMPOTENT MATRICES 707are lin
- Page 709 and 710: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 711 and 712: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 713 and 714: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 715 and 716: E.4. ALGEBRA OF PROJECTION ON AFFIN
- Page 717 and 718: E.5. PROJECTION EXAMPLES 717a ∗ 2
- Page 719 and 720: E.5. PROJECTION EXAMPLES 719where Y
- Page 721 and 722: E.5. PROJECTION EXAMPLES 721(B.4.2)
- Page 723 and 724: E.6. VECTORIZATION INTERPRETATION,
- Page 725 and 726: E.6. VECTORIZATION INTERPRETATION,
- Page 727 and 728: E.6. VECTORIZATION INTERPRETATION,
- Page 729 and 730: E.7. PROJECTION ON MATRIX SUBSPACES
- Page 731 and 732: E.7. PROJECTION ON MATRIX SUBSPACES
- Page 733 and 734: E.8. RANGE/ROWSPACE INTERPRETATION
- Page 735 and 736: E.9. PROJECTION ON CONVEX SET 735As
- Page 737 and 738: E.9. PROJECTION ON CONVEX SET 737Wi
- Page 739 and 740: E.9. PROJECTION ON CONVEX SET 739R(
- Page 741 and 742: E.9. PROJECTION ON CONVEX SET 741E.
- Page 743 and 744: E.9. PROJECTION ON CONVEX SET 743E.
- Page 745 and 746: E.9. PROJECTION ON CONVEX SET 745Un
D.2. TABLES OF GRADIENTS AND DERIVATIVES 697D.2.7exponentialMatrix exponential. [77,3.6,4.5] [331,5.4]∇ X e tr(Y T X) = ∇ X dete Y TX = e tr(Y T X) Y (∀X,Y )∇ X tre Y X = e Y T X T Y T = Y T e XT Y T∇ x 1 T e Ax = A T e Ax∇ x 1 T e |Ax| = A T δ(sgn(Ax))e |Ax| (Ax) i ≠ 0∇ x log(1 T e x ) = 11 T e x ex∇ 2 x log(1 T e x ) = 11 T e x (δ(e x ) − 11 T e x ex e xT )k∏∇ x x 1 ki = 1 (∏ kx 1 ki=1 k i)1/xi=1∇ 2 xk∏x 1 ki = − 1 ( k)(∏x 1 ki=1 k i δ(x) −2 − 1 )i=1 k (1/x)(1/x)Tddt etY = e tY Y = Y e tYddt eX+ t Y = e X+ t Y Y = Y e X+ t Y ,d 2dt 2 e X+ t Y = e X+ t Y Y 2 = Y e X+ t Y Y = Y 2 e X+ t Y ,XY = Y XXY = Y Xd jdt j e tr(X+ t Y ) = e tr(X+ t Y ) tr j (Y )
Change language