Matrix Binomials ?

1. Feb 24, 2008

clarex

Troubles interpreting what this wants from me.

X = (1 1
1 1)

Y = (1 -1
-1 1)

Let A = aX and B = bY, where a and b are constants.

Now consider M= (a+b a-b
a-b a+b)

Find THE general statement that expresses M^n in terms of aX and bY.

I'm completely lost. Can anyone direct me in the right direction?

2. Feb 25, 2008

Mystic998

Okay, so I'm sure you noticed that M = aX + bY. Then M^n = (aX + bY)^n, right? Now, if X and Y were just plain old real numbers (or variables if you like), what would you do to expand (aX + bY)^n

3. Feb 25, 2008

HallsofIvy

Especially since you titled this matrix binomials !

4. Feb 27, 2008

John Creighto

Diagonalize the matrix and then take the nth power of the diagonal matrix. You can generally extend scalar functions to matrix functions in this manner provided the eignvalues are unique and the eigenvectors or orthogonal. If you want once you have taken the nth power of the diagonal matrix you can apply the binomial theorm to expand the binomial terms on the diagonal.

5. Feb 27, 2008

HallsofIvy

My interpretation of "Find THE general statement that expresses M^n in terms of aX and bY" is that (aX+ bY)n would be sufficient.