Let I be the identity matrix, and let N be a nilpotent matrix order k. Then the matrix exponential function is defined as:
exp(N) := I + N + (1/2!)*N^2 + (1/3!)*N^3 + ... + (1/k!)*N^k
Similarly, we may define the matrix logarithm as the function
-log(N) := I - N + (I - N) + (I - N)^2 + (I - N)^3 ... + (I - N)^k
Show that exp(log(N)) = N.
The Attempt at a Solution
I'm not 100% sure, but I think the prof made a mistake. I can't seem to get this into an identity. He gave us a version of the logarithm function as:
log(N) = I - N + (I - N)/1! + (I - N)^2/2! + (I - N)^3/3! ... + (I - N)^k/k!
And I had trouble getting an identity out of it. I was going to point out this mistake, but he corrected himself in class and gave us the updated version (without the factorial) and it still doesn't come out to an identity for me. When k = 2, I get
-log(exp(N)) = -2N
and when it's order 3, I get
-log(exp(N)) = -2N - (1/3)*N^3
So I'm feeling certain it's still not right. If I'm not mistaken, it supposed to be a generalization of the logarithmic function for real numbers:
So I might try out this function and hand that in, telling him the corrected version just didn't work for me. My hunch is that the first term shouldn't be I - N but instead just I, since you'd want (I - N)^0/0! = I.
Can anyone else figure out what he really meant?