General Diagonalization Example

Process

1. Find the eigenvalues
2. Find the algebraic multiplicity of each eigenvalue
3. Check if $dim(V)=m_1+\cdots +m_k$
4. Check each $dim(E_{{\lambda }_i})=m_i$

Example

Determine if

$$\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 2\\ 0& 0& 3\end{array}\right]$$

is diagonizable

1. Find eigenvalues

$$X(\lambda )=\det \left[\begin{array}{ccc}1-\lambda & 1& 0\\ 0& 1-\lambda & 2\\ 0& 0& 3-\lambda \end{array}\right]$$

$X(\lambda )=(1-\lambda {)}^2(3-\lambda )$ 2. Find Algebraic Multiplicity.

• $Alg(1)=2$
• $Alg(2)=1$

3. Then, we have: $dim(V)=3=2+1=Alg(1)+Alg(3)$ (This is the condition of general diagonizability)
4. Then, we check $Alg(1)=Geo(1)$ and $Alg(3)=Geo(3)$
5. Compute $E_1=ker(T-1I)$

$$⟺\left[\begin{array}{cccc}1-1& 1& 0& 0\\ 0& 1-1& 2& 0\\ 0& 0& 3-1& 0\end{array}\right]\stackrel{EF}{\to }\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$$

6. Then, $E_1=span\{(1,0,0)\}$
7. This gives $Geo(1)=dim(E_1)=1$
8. Note that $Geo(1)=1,Alg(1)=2$. It fails the condition of general diagonizable