![[Graphics:Images/EigenvaluesMod_gr_180.gif]](../Images/EigenvaluesMod_gr_180.gif){kind=small}
.Example 4. Find the
eigenvalues and eigenvectors of the matrix .
Solution 4.
Find the characteristic polynomial and the eigenvalues.
![[Graphics:../Images/EigenvaluesMod_gr_181.gif]](../Images/EigenvaluesMod_gr_181.gif)
![[Graphics:../Images/EigenvaluesMod_gr_182.gif]](../Images/EigenvaluesMod_gr_182.gif)
Investigate the eigen-pairs ![[Graphics:../Images/EigenvaluesMod_gr_183.gif]](../Images/EigenvaluesMod_gr_183.gif){kind=small}
and ![[Graphics:../Images/EigenvaluesMod_gr_184.gif]](../Images/EigenvaluesMod_gr_184.gif){kind=small}
.
Introduce the free variables and find the eigenvector.
![[Graphics:../Images/EigenvaluesMod_gr_185.gif]](../Images/EigenvaluesMod_gr_185.gif)
![[Graphics:../Images/EigenvaluesMod_gr_186.gif]](../Images/EigenvaluesMod_gr_186.gif)
![[Graphics:../Images/EigenvaluesMod_gr_187.gif]](../Images/EigenvaluesMod_gr_187.gif)
![[Graphics:../Images/EigenvaluesMod_gr_188.gif]](../Images/EigenvaluesMod_gr_188.gif)
The eigenvalue is repeated, and there are two linearly independent eigenvectors.
Investigate the eigen-pairs ![[Graphics:../Images/EigenvaluesMod_gr_189.gif]](../Images/EigenvaluesMod_gr_189.gif){kind=small}
and ![[Graphics:../Images/EigenvaluesMod_gr_190.gif]](../Images/EigenvaluesMod_gr_190.gif){kind=small}
.
For ![[Graphics:../Images/EigenvaluesMod_gr_193.gif]](../Images/EigenvaluesMod_gr_193.gif){kind=small}
,
set s=0 in W and
get
Verify the eigenpair.
For ![[Graphics:../Images/EigenvaluesMod_gr_198.gif]](../Images/EigenvaluesMod_gr_198.gif){kind=small}
,
set t=0 in W and
get
Verify the eigenpair.
Investigate the eigen-pair ![[Graphics:../Images/EigenvaluesMod_gr_203.gif]](../Images/EigenvaluesMod_gr_203.gif){kind=small}
Introduce the free variables and find the eigenvector.
![[Graphics:../Images/EigenvaluesMod_gr_191.gif]](../Images/EigenvaluesMod_gr_191.gif){kind=small}
![[Graphics:../Images/EigenvaluesMod_gr_192.gif]](../Images/EigenvaluesMod_gr_192.gif){kind=small}
![[Graphics:../Images/EigenvaluesMod_gr_194.gif]](../Images/EigenvaluesMod_gr_194.gif)
![[Graphics:../Images/EigenvaluesMod_gr_195.gif]](../Images/EigenvaluesMod_gr_195.gif){kind=small}
![[Graphics:../Images/EigenvaluesMod_gr_196.gif]](../Images/EigenvaluesMod_gr_196.gif)
![[Graphics:../Images/EigenvaluesMod_gr_197.gif]](../Images/EigenvaluesMod_gr_197.gif)
![[Graphics:../Images/EigenvaluesMod_gr_199.gif]](../Images/EigenvaluesMod_gr_199.gif)
![[Graphics:../Images/EigenvaluesMod_gr_200.gif]](../Images/EigenvaluesMod_gr_200.gif){kind=small}
![[Graphics:../Images/EigenvaluesMod_gr_201.gif]](../Images/EigenvaluesMod_gr_201.gif)
![[Graphics:../Images/EigenvaluesMod_gr_202.gif]](../Images/EigenvaluesMod_gr_202.gif)
![[Graphics:../Images/EigenvaluesMod_gr_204.gif]](../Images/EigenvaluesMod_gr_204.gif)
![[Graphics:../Images/EigenvaluesMod_gr_205.gif]](../Images/EigenvaluesMod_gr_205.gif)
![[Graphics:../Images/EigenvaluesMod_gr_206.gif]](../Images/EigenvaluesMod_gr_206.gif)
![[Graphics:../Images/EigenvaluesMod_gr_207.gif]](../Images/EigenvaluesMod_gr_207.gif)
Verify the eigenpair.
It was good fortune that the three eigenvectors are linearly independent.
We can compare this with the results obtained using Mathematicas Eigensystem procedure.
(c) John H. Mathews 2004
![[Graphics:../Images/EigenvaluesMod_gr_208.gif]](../Images/EigenvaluesMod_gr_208.gif)
![[Graphics:../Images/EigenvaluesMod_gr_209.gif]](../Images/EigenvaluesMod_gr_209.gif)
![[Graphics:../Images/EigenvaluesMod_gr_210.gif]](../Images/EigenvaluesMod_gr_210.gif)
![[Graphics:../Images/EigenvaluesMod_gr_211.gif]](../Images/EigenvaluesMod_gr_211.gif)
![[Graphics:../Images/EigenvaluesMod_gr_212.gif]](../Images/EigenvaluesMod_gr_212.gif)
![[Graphics:../Images/EigenvaluesMod_gr_213.gif]](../Images/EigenvaluesMod_gr_213.gif)