Section 32.5 Examples
Example 32.5.1. Putting a nilpotent matrix into elementary nilpotent form.
Here we will apply Procedure 32.4.2 to the matrix
A=[120−137−1−4−19−2−4610−1−6].
Compute the powers of A:
A2=[16−1−316−1−343−1−3113−2−6],A3=[25−1−325−1−3−4−1026615−3−9],A4=0.
Since every column of A3 is nonzero, we can choose v to be any of the standard basis vectors. Let's choose v=e1. Then, using
p1=v=[1000],p2=Av=[13−16],p3=A2v=[1141],p4=A3v=[22−46],
we get
P=[111203120−14−40616].
Because of our choice of v=e1, there wasn't any need to actually compute the products Ajv to obtain the columns of P, as Aje1 is just the first column of Aj.
There also isn't any need to compute P−1AP, as we know our procedure will result in elementary nilpotent form
P−1AP=[0000100001000010].