Matrix multiplication 1

Question

I want to ask about the equation i hv marked a question mark.

(p^-1qp)ⁿ=p^-1qⁿp  how??

Why is there no power on matrix p ?

Answer 1

(p^-1qp)ⁿ=p^-1qⁿp 

To understand why is so happening,

lets take value of n = 2  

(p^-1qp)²           =(p^-1qp) (p^-1qp)                   

= p^-1 q p p^-1 q p             ∵  ( AB)C = A ( BC)

= p^-1 q I_n q p                 ∵   AA^-1 =  I_n

= p^-1 q  q p  

=  p^-1 q² p 

Similarly you can check and assure yourself with other values of n.

So (p^-1qp)ⁿ=p^-1qⁿp 

PS : One of the most important use of matrix diagonalization is calculating higher powers of a square matrix.

Ref : http://www.maths.lse.ac.uk/personal/martin/fme4a.pdf and https://en.wikipedia.org/wiki/Diagonalizable_matrix#An_application