Centralizer in GL(2,R)

Question

Given $$GL(2,\Bbb{R})$$ under the operation of matrix multiplication:

What is the centralizer of the diagonal matrix with $2, 3$ along its main diagonals in that order?

What is the centralizer in $GL(2,\Bbb{R})$ of the matrix $$\begin{pmatrix} 1 & 1\\ 0 & 1\\ \end{pmatrix} $$

I know that the identity is part of both centralizer sets but I am unable to think of any other elements that would commute with either matrices

Answer 1

1) You are looking for a matrix $\bigl(\begin{smallmatrix} a&b \\ c&d \end{smallmatrix} \bigr)$ which satisfies:
$\bigl(\begin{smallmatrix} a&b \\ c&d \end{smallmatrix} \bigr)\bigl(\begin{smallmatrix} 2&0 \\ 0&3 \end{smallmatrix} \bigr)=\bigl(\begin{smallmatrix} 2&0 \\ 0&3 \end{smallmatrix} \bigr)\bigl(\begin{smallmatrix} a&b \\ c&d \end{smallmatrix} \bigr)$
If you make the calculation you will find the following:
$2c=3c$ so that $c=0$ and $2b=3b$ so that $b=0$
Therefore the answer is $\bigl(\begin{smallmatrix} a&0 \\ 0&d \end{smallmatrix} \bigr)$ with $a \neq 0$ and $d \neq 0$ (because they are in $GL(2, \mathbb{R})$)
2) With a similar argument and calculation, you will find $\bigl(\begin{smallmatrix} a&b \\ 0&a \end{smallmatrix} \bigr)$ with $a \neq 0$

Answer 2

Hint: Commuting operators stabilize each other's eigenspaces - if $ Av = \lambda v $ for some scalar $ \lambda $, then $ A(Bv) = B(Av) = B(\lambda v) = \lambda(Bv) $.