Example 2
Example 2MatrixForm[B]
Out[35]=
2 -1 2 0 0 3 -1 0 0 1 1 0 0 1 -3 5
The following finds the characteristic polynomial of B.
In[36]:=
Det[B-x IdentityMatrix[4]]
Out[36]=
2 3 4
40 - 68 x + 42 x - 11 x + x
In[37]:=
Factor[Det[B-x IdentityMatrix[4]]]
Out[37]=
3
(-5 + x) (-2 + x)
Some shortcuts.
In[38]:=
Bm2I=B-2IdentityMatrix[4];
Bm5I=B-5IdentityMatrix[4];
Let's check (-5+x)(-2+x).
In[39]:=
MatrixForm[Bm5I.Bm2I]
Out[39]=
0 4 -7 0 0 -3 3 0 0 -3 3 0 0 -2 2 0
Let's check (-5+x)(-2+x)^2.
In[40]:=
MatrixForm[Bm5I.Bm2I.Bm2I]
Out[40]=
0 -3 3 0 0 0 0 0 0 0 0 0 0 0 0 0
By default (and the Cayley-Hamilton Theorem), the minimal polynomial is the
characteristic polynomial (suitably written to be monic):
(x-2)^3 (x-5)
Recall the Jordan form of B.
In[41]:=
MatrixForm[JB]
Out[41]=
2 1 0 0 0 2 1 0 0 0 2 0 0 0 0 5
Compare the size of the Jordan block for each eigenvalue with the multiplicity of that eigenvalue in the MINIMAL polynomial.