# Matrix Diagonalization

1. Mar 21, 2009

1. The problem statement, all variables and given/known data
My linear algebra textbook defines...

similar matrices: A = C^-1BC
diagonalized similar matrices: A = CDC^-1
A^n = C^-1*D^n*C

Why do the C^-1 and C's get switched around between the definitions? Doesn't order of multiplication matter? Are these the correct definitions? Is A^n really the opposite of the definition for diagonalized similar matrices, or is this an error?

2. Relevant equations

3. The attempt at a solution

2. Mar 21, 2009

### Hurkyl

Staff Emeritus
That couldn't possibly have been the definition your book gave. What did you omit?

3. Mar 21, 2009

Nope, that's it! He has us using his own lecture notes, instead of a published textbook. This is why I thought it could potentially be an error.

So if I'm diagonalizing a matrix, there exists C such that A = CDC^-1 but when I want to find A^n, I solve C^-1*D^n*C (the C inverse and C switch spots). Is that on purpose?

4. Mar 21, 2009

### Hurkyl

Staff Emeritus
Nothing else, really? Nothing like
A and B are similar matrices if ...​
or
... if there exists an invertible matrix C such that ...​
?

P.S. if $A = C D C^{-1}$, then we do indeed have $A^n = C D^n C^{-1}$. (For nonnegative integers n. Negative is allowed if D is invertible) And, of course, if we have the equation $A = C^{-1} D C$ then we can infer $A^n = C^{-1} D^n C$