6.6. EXERCISES 149

16. ↑Now in the above situation show that there is a basis for V = ker(N p) such that withrespect to this basis, the matrix of N is block diagonal of the form

B1

B2. . .

Br

 (6.4)

where the size of the blocks is decreasing from upper left to lower right and eachblock is of the form given in 6.3. Hint: Repeat the argument leading to this equationfor each β yi

where the ordered basis for ker(N p) is of the form{β y1

,β y2, · · · ,β yr

}arranged so that the length of β yi

is at least as long as the length of β yi+1.

17. ↑Now suppose the minimum polynomial for A ∈L (V,V ) is

p(λ ) =r

∏i=1

(λ −µ i)mi

Thus from what was shown above,

V =r⊕

i=1

ker((A−µ iI)mi)≡

r⊕i=1

ker(Nmi

i

)where Ni is the restriction of (A−µ iI) to Vi ≡ ker((A−µ iI)

mi). Explain why thereare ordered bases β 1, · · · ,β r, β j being a basis for Vj such that with respect to thisbasis, the matrix of Ni has the form

B1

B2. . .

Bsi

each Bk having ones down the super diagonal and zeros elsewhere. Now explain whyeach Vi is A invariant and the basis just described yields a matrix for A which is ofthe form 

J1. . .

Jr

where

Jk =

J1 (µk)

. . .

Jsk (µk)

