![[Graphics:Images/EigenvaluesMod_gr_289.gif]](../Images/EigenvaluesMod_gr_289.gif)
. Example 7. Find the
eigenvalues and eigenvectors of the matrix .
Solution 7.
Find the characteristic polynomial and the eigenvalues.
![[Graphics:../Images/EigenvaluesMod_gr_290.gif]](../Images/EigenvaluesMod_gr_290.gif)
![[Graphics:../Images/EigenvaluesMod_gr_291.gif]](../Images/EigenvaluesMod_gr_291.gif)
Investigate the eigen-pair ![[Graphics:../Images/EigenvaluesMod_gr_292.gif]](../Images/EigenvaluesMod_gr_292.gif){kind=small}
Introduce the free variables and find the eigenvector.
![[Graphics:../Images/EigenvaluesMod_gr_293.gif]](../Images/EigenvaluesMod_gr_293.gif)
![[Graphics:../Images/EigenvaluesMod_gr_294.gif]](../Images/EigenvaluesMod_gr_294.gif)
![[Graphics:../Images/EigenvaluesMod_gr_295.gif]](../Images/EigenvaluesMod_gr_295.gif)
![[Graphics:../Images/EigenvaluesMod_gr_296.gif]](../Images/EigenvaluesMod_gr_296.gif)
Verify the eigenpair.
Investigate the eigen-pair ![[Graphics:../Images/EigenvaluesMod_gr_299.gif]](../Images/EigenvaluesMod_gr_299.gif){kind=small}
Introduce the free variables and find the eigenvector.
![[Graphics:../Images/EigenvaluesMod_gr_297.gif]](../Images/EigenvaluesMod_gr_297.gif)
![[Graphics:../Images/EigenvaluesMod_gr_298.gif]](../Images/EigenvaluesMod_gr_298.gif)
![[Graphics:../Images/EigenvaluesMod_gr_300.gif]](../Images/EigenvaluesMod_gr_300.gif)
![[Graphics:../Images/EigenvaluesMod_gr_301.gif]](../Images/EigenvaluesMod_gr_301.gif)
![[Graphics:../Images/EigenvaluesMod_gr_302.gif]](../Images/EigenvaluesMod_gr_302.gif)
![[Graphics:../Images/EigenvaluesMod_gr_303.gif]](../Images/EigenvaluesMod_gr_303.gif)
Verify the eigenpair.
Investigate the eigen-pair ![[Graphics:../Images/EigenvaluesMod_gr_306.gif]](../Images/EigenvaluesMod_gr_306.gif){kind=small}
Introduce the free variables and find the eigenvector.
![[Graphics:../Images/EigenvaluesMod_gr_304.gif]](../Images/EigenvaluesMod_gr_304.gif)
![[Graphics:../Images/EigenvaluesMod_gr_305.gif]](../Images/EigenvaluesMod_gr_305.gif)
![[Graphics:../Images/EigenvaluesMod_gr_307.gif]](../Images/EigenvaluesMod_gr_307.gif)
![[Graphics:../Images/EigenvaluesMod_gr_308.gif]](../Images/EigenvaluesMod_gr_308.gif)
![[Graphics:../Images/EigenvaluesMod_gr_309.gif]](../Images/EigenvaluesMod_gr_309.gif)
![[Graphics:../Images/EigenvaluesMod_gr_310.gif]](../Images/EigenvaluesMod_gr_310.gif)
Verify the eigenpair.
Investigate the eigen-pair ![[Graphics:../Images/EigenvaluesMod_gr_313.gif]](../Images/EigenvaluesMod_gr_313.gif){kind=small}
Introduce the free variables and find the eigenvector.
![[Graphics:../Images/EigenvaluesMod_gr_311.gif]](../Images/EigenvaluesMod_gr_311.gif)
![[Graphics:../Images/EigenvaluesMod_gr_312.gif]](../Images/EigenvaluesMod_gr_312.gif)
![[Graphics:../Images/EigenvaluesMod_gr_314.gif]](../Images/EigenvaluesMod_gr_314.gif)
![[Graphics:../Images/EigenvaluesMod_gr_315.gif]](../Images/EigenvaluesMod_gr_315.gif)
![[Graphics:../Images/EigenvaluesMod_gr_316.gif]](../Images/EigenvaluesMod_gr_316.gif)
![[Graphics:../Images/EigenvaluesMod_gr_317.gif]](../Images/EigenvaluesMod_gr_317.gif)
Verify the eigenpair.
Investigate the eigen-pair ![[Graphics:../Images/EigenvaluesMod_gr_320.gif]](../Images/EigenvaluesMod_gr_320.gif){kind=small}
Introduce the free variables and find the eigenvector.
![[Graphics:../Images/EigenvaluesMod_gr_318.gif]](../Images/EigenvaluesMod_gr_318.gif)
![[Graphics:../Images/EigenvaluesMod_gr_319.gif]](../Images/EigenvaluesMod_gr_319.gif)
![[Graphics:../Images/EigenvaluesMod_gr_321.gif]](../Images/EigenvaluesMod_gr_321.gif)
![[Graphics:../Images/EigenvaluesMod_gr_322.gif]](../Images/EigenvaluesMod_gr_322.gif)
![[Graphics:../Images/EigenvaluesMod_gr_323.gif]](../Images/EigenvaluesMod_gr_323.gif)
![[Graphics:../Images/EigenvaluesMod_gr_324.gif]](../Images/EigenvaluesMod_gr_324.gif)
Verify the eigenpair.
The five eigen-pairs are:
We can compare this with the results obtained using Mathematicas Eigensystem procedure.
(c) John H. Mathews 2004
![[Graphics:../Images/EigenvaluesMod_gr_325.gif]](../Images/EigenvaluesMod_gr_325.gif)
![[Graphics:../Images/EigenvaluesMod_gr_326.gif]](../Images/EigenvaluesMod_gr_326.gif)
![[Graphics:../Images/EigenvaluesMod_gr_327.gif]](../Images/EigenvaluesMod_gr_327.gif)
![[Graphics:../Images/EigenvaluesMod_gr_328.gif]](../Images/EigenvaluesMod_gr_328.gif)
![[Graphics:../Images/EigenvaluesMod_gr_329.gif]](../Images/EigenvaluesMod_gr_329.gif)
![[Graphics:../Images/EigenvaluesMod_gr_330.gif]](../Images/EigenvaluesMod_gr_330.gif)