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v2007.09.13 - Convex Optimization

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564 APPENDIX D. MATRIX CALCULUSwhich can be proved by substitution of variables in (1590).second-order total differential due to any Y ∈R K×L isThe mn thd 2 g mn (X)| dX→Y= ∑ i,j∑k,l∂ 2 g mn (X)Y kl Y ij = tr(∇ X tr ( ∇g mn (X) T Y ) )TY∂X kl ∂X ij(1592)= ∑ ∂g mn (X + ∆t Y ) − ∂g mn (X)limY ij∆t→0 ∂Xi,jij ∆t(1593)g mn (X + 2∆t Y ) − 2g mn (X + ∆t Y ) + g mn (X)= lim∆t→0∆t 2 (1594)= d2dt 2 ∣∣∣∣t=0g mn (X+ t Y ) (1595)Hence the second directional derivative,→Ydg 2 (X) ∆ =⎡⎢⎣⎡tr(∇tr ( ∇g 11 (X) T Y ) )TY =tr(∇tr ( ∇g 21 (X) T Y ) )TY ⎢⎣ .tr(∇tr ( ∇g M1 (X) T Y ) )TYd 2 g 11 (X) d 2 g 12 (X) · · · d 2 g 1N (X)d 2 g 21 (X) d 2 g 22 (X) · · · d 2 g 2N (X). .d 2 g M1 (X) d 2 g M2 (X) · · ·.d 2 g MN (X)⎤⎥∈ R M×N⎦∣∣dX→Ytr(∇tr ( ∇g 12 (X) T Y ) )TY · · · tr(∇tr ( ∇g 1N (X) T Y ) ) ⎤TYtr(∇tr ( ∇g 22 (X) T Y ) )TY · · · tr(∇tr ( ∇g 2N (X) T Y ) )T Y ..tr(∇tr ( ∇g M2 (X) T Y ) )TY · · · tr(∇tr ( ∇g MN (X) T Y ) ⎥) ⎦TY⎡=⎢⎣∑ ∑i,jk,l∑ ∑i,jk,l∂ 2 g 11 (X)∂X kl ∂X ijY kl Y ij∂ 2 g 21 (X)∂X kl ∂X ijY kl Y ij.∑ ∑∂ 2 g M1 (X)∂X kl ∂X ijY kl Y iji,jk,l∑∑∂ 2 g 12 (X)∂X kl ∂X ijY kl Y ij · · ·i,ji,jk,l∑∑∂ 2 g 22 (X)∂X kl ∂X ijY kl Y ij · · ·k,l.∑∑∂ 2 g M2 (X)∂X kl ∂X ijY kl Y ij · · ·i,jk,l∑∑⎤∂ 2 g 1N (X)∂X kl ∂X ijY kl Y iji,j k,l∑∑∂ 2 g 2N (X)∂X kl ∂X ijY kl Y iji,j k,l. ⎥∑ ∑ ∂ 2 g MN (X) ⎦∂X kl ∂X ijY kl Y iji,jk,l(1596)

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