16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 1

Elementary Canonical Forms

Mathematics — Linear Algebra · primary-decomposition-minimal-polynomial

Problem

Let

T

be a linear operator on

R^3

which is represented in the standard ordered basis by the matrix

\begin{bmatrix} 6 & -3 & -2 \\ 4 & -1 & -2 \\ 10 & -5 & -3 \end{bmatrix}.

Express the minimal polynomial

p

for

T

in the form

p = p_1p_2

, where

p_1

and

p_2

are monic and irreducible over the field of real numbers. Let

W_i

be the null space of

p_i(T)

. Find bases

\mathfrak{B}_i

for the spaces

W_1

and

W_2

. If

T_i

is the operator induced on

W_i

T

, find the matrix of

T_i

in the basis

\mathfrak{B}_i

(above).