2011-05-22, 01:13 AM
Is there a specific, clean formula that I can use to raise a matrix to an arbitrary power? To keep this simple, let's use square, 2x2 matrices.
What is the "geometric" interpretation of doing such? I know that matrix multiplication serves to linearly distort spaces by changing the basis vectors, so repeatedly doing this would be repeatedly distorting the basis vectors by a certain amount. Thus, would raising a matrix to an arbitrary power distort the basis vectors to some kind of vector in between? Would each of the entries vary continuously?
The reason I ask is that I recently found out what a matrix exponential is:
e^[A] = [I] + [A] + [A][A]/2! + [A][A][A]/3! + ...
However, if I wanted to compute 10^[A], then I would have to rewrite it in terms of e. I can do this two ways:
10^[A] = (e^ln(10))^[A] = (e^[A])^ln(10)
or:
10^[A] = (e^ln(10))^[A] = e^([A]*ln(10))
This implies that:
([I] + [A] + [A][A]/2! + [A][A][A]/3! + ...)^ln(10)
= [I] + [A]*ln(10) + [A][A]*ln(10)^2/2! + [A][A][A]*ln(10)^3/3! + ...
Which doesn't really make sense to me.
What is the "geometric" interpretation of doing such? I know that matrix multiplication serves to linearly distort spaces by changing the basis vectors, so repeatedly doing this would be repeatedly distorting the basis vectors by a certain amount. Thus, would raising a matrix to an arbitrary power distort the basis vectors to some kind of vector in between? Would each of the entries vary continuously?
The reason I ask is that I recently found out what a matrix exponential is:
e^[A] = [I] + [A] + [A][A]/2! + [A][A][A]/3! + ...
However, if I wanted to compute 10^[A], then I would have to rewrite it in terms of e. I can do this two ways:
10^[A] = (e^ln(10))^[A] = (e^[A])^ln(10)
or:
10^[A] = (e^ln(10))^[A] = e^([A]*ln(10))
This implies that:
([I] + [A] + [A][A]/2! + [A][A][A]/3! + ...)^ln(10)
= [I] + [A]*ln(10) + [A][A]*ln(10)^2/2! + [A][A][A]*ln(10)^3/3! + ...
Which doesn't really make sense to me.