# Linearizing dynamics - derivative of matrix wrt vector

alexjbuck
Hi,

I'm working on a project (not homework) where I have to linearize a set of non-linear equations of motion

They take the form:

x_dot(x,u) = B(x)*u + A(x)

where A and B are functions of state vector x. x_dot is a time derivative of x.
A - n x 1
B - n x m
u - m x 1

(wish I could get the latex to work... Its probably not broken, but rather I am...)

I also know that: (linearizing about some arbitrary x and u)

delta(x_dot) = (d(Bu)/dx + dA/dx)*delta x + (d(Bu)/du + dA/du)*delta u

I know dA/du = 0 and also that d(Bu)/du = B, but then the question is, how do I evaluate the first half? If I'm not mistaken it involves the derivative of a matrix with respect to a vector and I do not know how to go about this.

Thanks!

alexjbuck
I just realized that Bu is an n x m times a m x 1 thus the product is an n x 1 vector.

d(Bu)/dx is then just the derivative of a vector wrt a vector, which is the Jacobian if I'm not mistaken...

looks like I just answered my own question (I think)