# Matrices question

• Dec 12th 2006, 04:41 PM
rIBBON:lacedx
Matrices question
Hi everyone,

I'm writing a paper for my math class that is due really soon, but there's one thing I can't explain in my paper that I need to: why can't matrices be raised to a fraction exponent? I know that it won't work, but I don't know why and cannot find the answer why.

Is it because the denominator of the fractional exponent technically becomes the index when finding the root of a matrix, and you can't take the root of a matrix?

I'm dealing with square matrices, and I know that you can take the square root. But is that the only root you can take?

I'm so lost here. :(

Thank you for any help.
• Dec 12th 2006, 04:53 PM
topsquark
Quote:

Originally Posted by rIBBON:lacedx
Hi everyone,

I'm writing a paper for my math class that is due really soon, but there's one thing I can't explain in my paper that I need to: why can't matrices be raised to a fraction exponent? I know that it won't work, but I don't know why and cannot find the answer why.

Is it because the denominator of the fractional exponent technically becomes the index when finding the root of a matrix, and you can't take the root of a matrix?

I'm dealing with square matrices, and I know that you can take the square root. But is that the only root you can take?

I'm so lost here. :(

Thank you for any help.

Why ever not? I happen to know that a cube root of
-5 -6
2 -3

is

1 3
-1 0

As you can verify for yourself by multiplying it out. Finding the cube root of an arbitrary matrix, I will admit, would be a chore, but I suspect possible. You would need to be able to solve the following system:
Given a matrix
p q
r s

has the cube root
a b
c d

where
p = a^3 + 2abc + bcd
q = (a^2 + ad + bc + d^2)b
r = (a^2 + ad + bc + d^2)c
s = d^3 + 2bcd + abc

-Dan
• Dec 13th 2006, 03:53 AM
Fredrik
Quote:

Originally Posted by rIBBON:lacedx
...there's one thing I can't explain in my paper that I need to: why can't matrices be raised to a fraction exponent? I know that it won't work, but I don't know why and cannot find the answer why.

The problem isn't that it can't be done. The problem is that the function that takes a matrix X to X^(n/m) is multi-valued.

For example, consider the matrix equation A^2=1, where the "1" on the right hand side is the 2x2 unit matrix. Any solution to this equation could be denoted "1^(1/2)", but there's more than one solution. So there's no way to define "1^(1/2)" so that it's just one matrix. It would have to be a set containing several matrices.

One of the solutions is of course the unit matrix. The others are known as the Pauli spin matrices.