Matrix Binomials: Struggling to Interpret Question

[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?

 

[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

 

[HallsofIvy]

Especially since you titled this matrix binomials !

 

[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 theorem to expand the binomial terms on the diagonal.

 

[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.