Question

If eigenvalues are not distinct, then the matrix is still diagonalizable?

Answer 1

Answer:

Sometimes yes, sometimes no.

Step-by-step explanation:

Consider a couple examples.

$\displaystyle\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$

• has eigenvalues that are not distinct (1 and 1)
• is diagonalizable (it's already diagonal, so of course it's diagonalizable!)

However

$\displaystyle\left(\begin{array}{cc}1&1\\0&1\end{array}\right)$

• has eigenvalues that are not distinct (1 and 1)
• is not diagonalizable