I like examples.
| diagonalizable | not diagonalizable | |
|---|---|---|
| invertible | I_n, ![\left[\begin{array}{ccc} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 1\end{array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+3+%26+0+%26+0+%5C%5C+0+%26+-2+%26+0+%5C%5C+0+%26+0+%26+1%5Cend%7Barray%7D%5Cright%5D&bg=ffffff&fg=000&s=0&c=20201002) |
rotation about non-axis line by, say, 60 degrees in R^3, rotation 90 degrees in R^2: ![\left[\begin{array}{cc} 0 & -1 \\ 1 & 0\end{array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D+0+%26+-1+%5C%5C+1+%26+0%5Cend%7Barray%7D%5Cright%5D&bg=ffffff&fg=000&s=0&c=20201002) |
| not invertible | zero matrix, ![\left[\begin{array}{ccc} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0\end{array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+3+%26+0+%26+0+%5C%5C+0+%26+2+%26+0+%5C%5C+0+%26+0+%26+0%5Cend%7Barray%7D%5Cright%5D&bg=ffffff&fg=000&s=0&c=20201002) |
rotation 90 degrees plus projection onto yz-plane in R^3: ![\left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+0+%26+0+%26+0+%5C%5C+0+%26+0+%26+-1+%5C%5C+0+%26+1+%26+0%5Cend%7Barray%7D%5Cright%5D&bg=ffffff&fg=000&s=0&c=20201002) |
trying to be a mile wide AND a mile deep
I like examples.
| diagonalizable | not diagonalizable | |
|---|---|---|
| invertible | I_n, |
rotation about non-axis line by, say, 60 degrees in R^3, rotation 90 degrees in R^2: |
| not invertible | zero matrix, |
rotation 90 degrees plus projection onto yz-plane in R^3: |