# Exponential of a (defective) matrix

I have the following matrix:

$$M=\begin{bmatrix}a+c&e&c& e\\e& b+d &e &d\\-c &-e& -(b+c)& -e\\-e &-d& -e& -(a+d)\end{bmatrix}$$

I want to calculate the exponential of M. I believe M is a defective Matrix. I tried several methods, splitting, Sylvester Formula, etc. I tried to compute the Jordan form as is suggested in one post in Stackexchange, by using the jordan function of matlab but it does not work since it gives me just the matrix of eigenvectors and eigenvalues. Since:

$$[V,D]=eig(M)$$ $$e^M=V e^{diag(D)} V^{-1}$$ but the problem is that it is not possible to calculate the inverse of V.

Anyone has any idea what is the proper method to calculate this? Thanks in advance.

• Welcome to math.SE. You could use MathJax to prettify your formula – Yujie Zha May 23 '17 at 13:21
• The invertibility of $M$ depends on $a,b,c,d,e$. For $a=0, b=0$, it is not invertible. But if $a,b \neq 0$, then $a,b,c,d,e$ need to have a specific relationship to not be invertible ($e = \sqrt{ab+ac+bc}\sqrt{ab+ad+bd}/|a+b|$) or something. MatrixExp[{{a + c, e, c, e}, {e, b + d, e, d}, {-c, -e, -(b + c), -e}, {-e, -d, -e, -(a + d)}}] in Mathematica gives something nasty. – Batman May 23 '17 at 13:49
• @batman thanks i could also obtain some result with Mathematica, that i was not capable to obtain before. In fact the letters a,b,c,d,e mean other expressions that i didn't want to reveal. With that expressions (and i dont know why) i was not able to obtain any result with Mathematica. I wonder if it is possible by other algorithm to obtain simpler expressions. – Vitor Ribeiro May 24 '17 at 10:24