Question

How does the power of a matrix operate? If A is a matrix, [\A^{2}\] doesn't look like literally taking the powers of each entries in the matrix.

Answer 1

For instance, \[A=\begin{bmatrix} 0 & 0 & 3\\ 1 & 0 & 2\\ 0 & 5 & 0 \end{bmatrix}\]
\[A^{2} \neq \begin{bmatrix} 0 & 0 & 9\\ 1 & 0 & 4\\ 0 & 25 & 0 \end{bmatrix}\]

But rather, when calculated from a calculator, it is:
\[A^{2} = \begin{bmatrix} 0 & 15 & 0\\ 0 & 10 & 3\\ 5 & 0 & 10 \end{bmatrix}\]

How are these values obtained?

Answer 2

let's start with 2x2 matrices:

\[\left[\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right]\left[\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right]\] = \[\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\]
a=1*1+2*3
b=1*2+2*4
c=3*1+4*3
d=3*2+4*4

so you get \[\left[\begin{matrix}7 & 10 \\ 15 & 22\end{matrix}\right]\]

in general to get the number i j calcute the dot product of row j with column i. This rule can also be applied to 3x3 matrices. To get 15 in your example: i=2,j=1 so we take the dot product of (0,0,3) with (0,0,5) = 15

I hope this is a bit clear.

Answer 3

http://en.wikipedia.org/wiki/Matrix_multiplication#Powers_of_matrices

Answer 4

If this was a rectangular matrix, would this still work? Because I think if I had say 3x5 matrix, then there isn't row 4 and row 5 to carry out the operation, wouldn't it?

Answer 5

You can only take powers of square matrices, in general you multiply an mxn matrix only with an nxp matrix for any p. so the number of rows of the first matrix must be the same as the number of columns of the second matrix.

Answer 6

http://en.wikipedia.org/wiki/Matrix_decomposition

Answer 7

Thanks Thomas for the help and estudier for the links! thanks a lot! :)