My Math Forum Partial derivative of a matrix product w.r.t. a vector

 Linear Algebra Linear Algebra Math Forum

 December 6th, 2015, 11:19 AM #1 Newbie   Joined: Dec 2015 From: Palo Alto , California Posts: 1 Thanks: 0 Partial derivative of a matrix product w.r.t. a vector Hi all, Let $f(\chi_i,\zeta_i) : \mathbb{R}^m \rightarrow \mathbb{R}^n$, $Q \in \mathbb{R}^{n \times n }$ e $\chi_i \in \mathbb{R}^n$. I have to calculate the Hessian $$\frac{\partial^2}{\partial \chi_2^2 } \left( (\chi_2 - \chi_1 - f(\chi_2, \zeta_2))^T Q (\chi_2 - \chi_1 - f(\chi_2, \zeta_2) \right)$$ Since the product $( (\chi_2 - \chi_1 - f(\chi_2, \zeta_2) )^T Q (\chi_2 - \chi_1 - f(\chi_2, \zeta_2) )$ is a constant the hessian must be a matrix $n \times \n$. Working out that expression I obtained: $$2 \left[ Q + f(\chi_2, \zeta_2)^T Q \frac{\partial^2 f(\chi_2, \zeta_2)}{\partial \chi_2^2} + \frac{\partial f(\chi_2, \zeta_2)}{\partial \chi_2}^T Q \frac{\partial f(\chi_2, \zeta_2)}{\partial \chi_2} + \chi_1 Q \frac{\partial f(\chi_2, \zeta_2)}{\partial \chi_2} - \left( \chi_2^T Q \frac{\partial^2 f(\chi_2,\zeta_2)}{\partial \chi_2^2} + \frac{\partial f(\chi_2, \zeta_2)}{\partial \chi_2} Q \right) \right]$$ But I do not know how to solve this product $$f(\chi_2, \zeta_2)^T Q \frac{\partial^2 f(\chi_2, \zeta_2)}{\partial \chi_2^2}$$ THe second derivative of $f$ is, as far as I understood, a third order tensor . Therefore I have a vector which left-multiplies a matrix which left-multiplies a cubix and from this I must get a matrix $n \times n$. How would you do it? I thought to mutiply each "layer" of the cubix with the matrix so to have a cubix, and then multiply the cubix with the vector and sum along the third dimension so as to obtain a matrix. Does that make sense? Thanks very much, Bruno

 Tags derivative, matrix, partial, product, vector, wrt

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post zamzar1992 Calculus 1 April 22nd, 2016 06:37 PM whitegreen Linear Algebra 1 June 9th, 2015 06:11 AM 71GA Algebra 1 June 3rd, 2012 01:20 PM JamesKirk Linear Algebra 6 March 18th, 2011 03:17 PM bluetrain Calculus 1 February 24th, 2010 05:15 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top