Sensitivity

It is months into the semester and Dr. Ferrar just recently dropped a bomb on the class about the implications of the inverse [A] matrix. We have been studying linear systems that can be described as:

[A][x] = [b],

where [A] is the parameters of the system, [x] and [b] are blah blah blah. One can solve for the desired unknowns [x] by multiplying inverse [A] by [b]:

[x] = [A]^-1[b]

I must admit, I am sick of writing out and describing this very practical relationship. However, a few days ago, Dr. Ferrar decided to impart some wisdom he has been holding on to for a very long time. And it has renewed my interest in the course.

[A]^-1 just so happens to detail the degree to which each input of [b] effects the system and alters the outcome. It describes the relationship strength between elements of the system. For more visual interpreters, the sensitivity can also be illustrated in Matlab similar to the following.

With knowledge of sensitivity, an engineer can optimize a system and take control of how that system responds. From the above sensitivity matrix, one can see that the top left (00) responds very strongly to input, while the bottom left and top right edges do not. Very fascinating stuff.

And oh, doing Cholesky Factorization by hand is a pain in the [A]^-1ss.