1. ## 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.

2. 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

3. 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.