Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Fractional Matrices
Name: Dhruv
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: N/A


Question:
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)?



Replies:
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 multiplication,

A * A  = ( 8   0)
              ( 0    8)

then the square-root matrix A could be any of

+/- 1   * ( 2    2)
               (  2  -2)

-OR-

+/-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.

best wishes,

Michael S. Pierce
Materials Science Division
Argonne National Laboratory



Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs

NEWTON AND ASK A SCIENTIST
Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory