Differentiating the matrix exponential with respect to a scalar

[nomaikai]

Homework Statement

Let's say A is a 7x7 matrix which is defined as [a b c 0 0 0 0; b a 0 d 0 0 0; c 0 a b e 0 0; 0 d b f 0 e 0; 0 0 d 0 f b g; 0 0 0 d b f h; 0 0 0 0 0 0 0] where semicolon (;) represent a new row and a space is a new column.

Homework Equations

If y = expm (A*t), where expm refers to the exponential matrix and t is time, how can we solve the differentiation of y with respect to one of the paramaters in A, let's say a.

dy/da = ??

The Attempt at a Solution

For a 2x2 matrix, Z, we know the expm (Z*t) = V*D*inv(V), where V is the eigenvector of Z and inv (V) refers to the inverse of V and D is a diagonal matrix with elements equal to the exponential of the eigenvalues of Z.

Then differentiate expm (Z*t) with respect to a parameter in Z can be calculated easily by computing V*D*inv(V) first, after that differentiate it with respect to the parameter. However, a 7x7 matrix will give you a hell lot of problems to find the eigenvectors and eigenvalues and multiple them together will make the equations to be extremely long. Since the equations are tedious, differentiating it with respect to a parameter of interest is difficult too.

Thus, I am interested to know is there any simple or easy way to solve this kind of problem which involves a large matrix. I came across a few matrix identities which show that d expm(A*t)/dt = A *expm(A*t). Hopefully, someone knows what the answer would be for my problem which is d expm (A*t)/da, where a is one of the elements in A. Thanks. Sorry for the long explanation and problem set up, please do not hesitate to ask me if you are not clear about any part of my question. Thanks. :)

 

[hunt_mat]

What you could do (with the help of a computer algebra package perhaps) but you could compute A^n, n=0,1,2,3,... and then compute the matrix exponential like that. After this, differentitate.

Another way would be to try and diagonalise A and then you can just compute the matrix expotential easily.