# How do I determine and explain if the following matrices are diagonalizable?

Determine if each of the following matrices is diagonalizable and explain why.

a)

$$\begin{bmatrix} 3 & 1 & 4 \\ 0 & 1 & 5 \\ 0 & 0 & 9 \end{bmatrix}$$

b)

$$\begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 5 & 6 & 8 \\ 3 & 6 & 7 & 9 \\ 4 & 8 & 9 & 1 \end{bmatrix}$$

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What, precisely, are you having trouble with? –  Hurkyl May 3 '13 at 23:09
Fomo Loco: you've been asking lots of questions, and getting many answers, and you keep coming back. Why, then, I wonder, if you're finding the site helpful, haven't you accepted ANY of the answers you've received? –  amWhy May 3 '13 at 23:21
Both are diagonalizable because of known facts: you know the eigenvalues of the first one and the second matrix has a very clear property. –  egreg May 3 '13 at 23:22
I am sorry amWhy. I wasn't doing it on purpose. How do I accept answers? –  Fomo Loco May 3 '13 at 23:45
You can also go back and accept earlier answers ;-) You get two reputation points for each answer you accept! –  amWhy May 4 '13 at 3:31

The matrix in b) has real elements and is symmetric ($A=A^T$). By the spectral theorem, we know that it is diagonalisable. (We even know that the eigenvalues are real, and that the eigenvectors can be chosen orthogonal.)