Kunze Linear Algebra · Problem 6.2.1

Elementary Canonical Forms

Mathematics — Methods · characteristic-values-matrices

Problem

In each of the following cases, let

T

be the linear operator on

R^2

which is represented by the matrix

A

in the standard ordered basis for

R^2

, and let

U

be the linear operator on

C^2

represented by

A

in the standard ordered basis. Find the characteristic polynomial for

T

and that for

U

, find the characteristic values of each operator, and for each such characteristic value

c

find a basis for the corresponding space of characteristic vectors.

A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad A = \begin{bmatrix} 2 & 3 \\ -1 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}.