$Post:=If[MatrixQ[#],MatrixForm[#],#]&

Exercise 3.4.1

A=Table[If[i>j,1,0],{i,4},{j,4}]

Id[4]=IdentityMatrix[4];
p=Det[x Id[4]-A]

lambda=x/.Solve[p==0,x]

S=NullSpace[lambda[[1]]Id[4]-A]

M=NullSpace[S]

P=Transpose[Join[S,M]]

T=Inverse[P].A.P

Exercise 3.4.2

A=Table[Mod[i+j,2],{i,4},{j,4}]

p=Det[x Id[4]-A]

lambda=x/.Solve[p==0,x]

Clear[S]
Do[S[i]=NullSpace[lambda[[i]]Id[4]-A],{i,4}]
{S[1],S[2],S[4]}

P=Transpose[Join[S[1],S[2],S[4]]]

T=Inverse[P].A.P

Exercise 3.4.3

A=Table[If[i<=2,2,0],{i,4},{j,4}]

p=Det[x Id[4]-A]

lambda=x/.Solve[p==0,x]

Do[S[i]=NullSpace[lambda[[i]]Id[4]-A],{i,4}]
{S[1],S[4]}

P=Transpose[Join[S[1],S[4]]]

T=Inverse[P].A.P

Exercise 3.4.4

A=Table[If[i==j,0,1],{i,4},{j,4}]

p=Det[x Id[4]-A]

lambda=x/.Solve[p==0,x]

Do[S[i]=NullSpace[lambda[[i]]Id[4]-A],{i,4}]
{S[1],S[4]}

P=Transpose[Join[S[1],S[4]]]

T=Inverse[P].A.P