Characteristic polynomial
This is the determinant of
```
 [–λ  0   2 ]
 [–3 1–λ  6 ]
 [ 0  0  1–λ]
```
which is (expand along the third row) (1-λ)²(–λ).
Eigenvalues
We can read off the roots. They are 0 and 1 (the root 1 has multiplicity 2). These are the eigenvalues.
Bases for eigenspaces
- λ=0
  So
```
 [–λ  0   2 ]
 [–3 1–λ  6 ]
 [ 0  0  1–λ]
```
  becomes
```
 [ 0 0 2]
 [–3 1 6]
 [ 0 0 1]
```
  Some row operations change this to
```
 [ 0 0 1]
 [–3 1 0]
 [ 0 0 0]
```
  which has rank 2. The null space is 1-dimensional. Surely x₃=0, and if x₁=1, then x₂=3. A basis for this eigenspace is the set consisting of the single vector
```
[1]
[3]
[0].
```
- λ=1
  So
```
 [–λ  0   2 ]
 [–3 1–λ  6 ]
 [ 0  0  1–λ]
```
  becomes
```
 [–1 0 2]
 [–3 0 6]
 [ 0 0 0]
```
  This matrix has rank 1. We can disregard the second row (it echos the first). There is no restriction on x₂, and x₁=2x₃. Thus x₂ and x₃ are free. This subspace has dimension 2. A basis for this eigenspace is the set consisting of the two vectors
```
[0] [2]
[1] [0]
[0] [1].
```

This matrix can be diagonalized -- there are enough linearly independent eigenvectors (two, the same as the multiplicity) in the eigenspace corresponding to λ=1: A=PDP^–1 where P is

 [1 0 2]
 [3 1 0]
 [0 0 1]

and D is

 [0 0 0]
 [0 1 0]
 [0 0 1].