I am wondering if it is possible to raise a
matrix to a fraction. Considering matrix A, is it possible to take A^(2/3)?
It is possible to perform that operation (or anything else you can
dream up) in terms of it being what is called a "scalar"
operation. Which means, in English, that you perform a particular
operation to each of the elements in a matrix. So you could raise
each number or variable inside the matrix to a particular power or
whatever you wish. However, I am not aware of any particular method
of applying that as an entire operation on a matrix. It would be
best to look for a text book on linear algebra. For the most part,
the introductory college linear algebra texts should be
accessible/readable to a highschool student with a little
effort. They usually do not require a knowledge of calculus or
things like that.
To "square" a matrix, A^2, could mean either squaring element in the
matrix separately or it could mean the matrix operation A * A with
the row-by-column multiplication and addition.
An interesting, related problem is that once you allow negative
numbers, matrices are no longer singularly defined by particular
operations. For instance, suppose that using the standard matrix
A * A = ( 8 0)
( 0 8)
then the square-root matrix A could be any of
+/- 1 * ( 2 2)
( 2 -2)
+/-1 * ( 2*sqrt(2) 0 )
( 0 2*sqrt(2) )
so that is 4 possible, unique solutions. The " +/- 1 *" above is
just a shorthand to say
multiply by plus or minus 1.
Anyways... That is all I can figure out at the moment. I do not
have a linear algebra text in front of me, but I am sure
mathematicians have fancy, well defined names and definitions for
all of the above.
Michael S. Pierce
Materials Science Division
Argonne National Laboratory
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Update: June 2012