I diagonal matrix is obviously diagonalizable since I can conjugate it with the identity. …(1)
Besides, a matrix 2×2 is diagonalizable iff it has two distinct eigenvalues….(2)
For example the matrix
$\begin{bmatrix}4&0\\0&4\end{bmatrix}$ has only one eigenvalue :4 of algebraic multiplicity 2,
then it shouldn't be diagonaliz
zable, should it? but it obviously is diagonalizable (because of (1)) What am doing wrong?
I am not very sure of (2), but in an exercise we were interested in characterizing the 2×2 non- diagonalizable matrices, and the professor said that the characteristic polynomial should have a double root, so only one eigenvalue of algebraic multiplicity 2, that's why I believed that to have instead a diagonalizable matrix, the eigenvalues should be distinct.
Best Answer
A typical 2 x 2 non-diagonalizable matrix is $$\pmatrix{ 1 & 1 \\ 0 & 1} $$ Its characteristic polynomial has one double-root, but its minimal polynomial is also $(x-1)^2$, which makes it different from the identity, whose char. poly has a double root, but whose minimal polyonomial is $(x-1)$.
What your prof. said was correct, but you negated it incorrectly. :)
By the way, I applaud your questioning this. Asking questions like this, even ones that seem stupid, is part of how you learn to recognize certain classes of errors and learn not to make them again. Go, you!