Transcription of Matrix Calculus - Stanford University
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Matrix Calculus - Stanford University
Appendix DMatrix CalculusFrom too much study, and from extreme passion, cometh madnesse. Isaac Newton[205, 5] Gradient, Directional derivative, Taylor GradientsGradientof a differentiable real functionf(x) :RK Rwith respect to its vectorargument is defined uniquely in terms of partial derivatives f(x), f(x) x1 f(x) f(x) xK RK(2053)while the second-order gradient of the twice differentiable real function with respect toitsvector argument is traditionally called theHessian; 2f(x), 2f(x) x21 2f(x) x1 x2 2f(x) x1 xK 2f(x) x2 x1 2f(x) x22 2f(x) x2 2f(x) xK x1 2f(x) xK x2 2f(x) x2K SK(2054)interpreted 2f(x) x1 x2= f(x) x1 x2= f(x) x2 x1= 2f(x) x2 x1(2055)Dattorro,Convex Optimization Euclidean Distance Geometry,M oo, 2005, D. Matrix CALCULUSThe gradient of vector-valued functionv(x) :R RNon real domain is a row vector v(x),h v1(x) x v2(x) x vN(x) xi RN(2056)while the second-order gradient is 2v(x),h 2v1(x) x2 2v2(x) x2 2vN(x) x2i RN(2057)Gradient of vector-valued functionh(x) :RK RNon vector domain is h(x), h1(x) x1 h2(x) x1 hN(x) x1 h1(x) x2 h2(x) x2 hN(x) h1(x) xK h2(x) xK hN(x) xK = [ h1(x) h2(x) hN(x) ] RK N(2058)while the second-order gradient has a three-dimensional written representation dubbedcubix; 2h(x).
Appendix D Matrix Calculus From too much study, and from extreme passion, cometh madnesse. −Isaac Newton [205, § 5] D.1 Gradient, Directional derivative, Taylor series D.1.1 Gradients Gradient of a differentiable real function f(x) : RK→R with respect to its vector argument is defined uniquely in terms of partial derivatives ∇f(x) ,
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