Diagonalizable and Invertible Functions

Question

In each of the following parts, either give an example of a linear function T: C$^2$ -> C$^2$ with the specified properties (and show that your example has the desired properties), or prove that no example exists.

(a) Diagonalizable and invertible. (b) Not diagonalizable and not invertible. (c) Diagonalizable, but not invertible. (d) Invertible, but not diagonalizable.

For (c) the matrix $\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}$is an example and for (d) the matrix $\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$is an example, but how do I get functions that give these matrices? For (a) and (b) I'm not sure of a matrix.

Answer 1

Unless otherwise stated, the function $T:\Bbb C^2\to\Bbb C^2$ associated to a matrix $$ A= \begin{bmatrix} a & b\\ c& d \end{bmatrix} $$ is the function $$ T(x,y)= A\begin{bmatrix} x\\ y\end{bmatrix} = \begin{bmatrix} a & b\\ c& d \end{bmatrix} \begin{bmatrix} x\\ y\end{bmatrix} = \begin{bmatrix} ax+by\\ cx+dy \end{bmatrix} $$

Answer 2

a) , I would consider the identity matrix..

b) , I would consider a nilpotent matrix: there is a k>1 : $N^k = 0$

Certainly not invertible, as for diagonalizable, you can suppose it is. Since N can only have 0 as an eigenvalue ( minimal polynomial divides $x^k$) , if it is diagonalizable then it's similar to the null matrix: $P^{-1}*N*P = 0$ => $N=0$