Canonical forms of linear operators in n-dimensional space

Canonical forms of linear operators in n-dimensional space


	Canonical forms of linear operators in n-dimensional space

The situation with canonical forms for linear operators in n-dimensional space is complicator than for linear operators on the plane. However, the steps of the procedure for finding the canonical forms remain unchanged. First, we calculate characteristic polynomial

$\begin{displaymath} \det(A - \lambda E ) = 0. \end{displaymath}$

Second, we find its roots-eigenvalues for A. Then the structure of the canonical form is dictated by the eigenvalues and the corresponding eigenvectors and adjoint vectors.

Here we present the building blocks for the canonical forms of linear operators. These blocks are arranged on the diagonal of the canonical forms.

For a single real eigenvalue

$\lambda_j$

the building block coincides with this eigenvalue and in the matrix C for the change of coordinates we put the corresponding eigenvector

$\xi_j.$

It is important to point out that the order of eigenvalues on the diagonal of the canonical form should match the order of the corresponding eigenvectors in C.

For a single complex eigenvalue

$\lambda_k = \alpha + i \beta$

we have the block

$\begin{displaymath} \left( \begin{array}{cc} \alpha & \beta\ -\beta & \alpha \end{array}\right) \end{displaymath}$

For a real eigenvalue

$\mu$

of a multiplicity m having just a single eigenvector

$\xi_j$

we have the Jordan block of the order m,

$\begin{displaymath} J_m(\mu)= \left( \begin{array}{ccccc} \mu & 1 & 0 & \dots ... ...ts & \mu & 1 \ 0 & 0 & \dots & 0 & \mu \end{array}\right). \end{displaymath}$

We put in the matrix C the columns

$\begin{displaymath} \xi_j, \xi_{j1}, \xi_{j2}, \dots , \xi_{j m-1}, \end{displaymath}$

where

$\xi_j$

is the eigenvector corresponding to

$\mu$

and

$\begin{displaymath} \xi_{j1}, \xi_{j2}, \dots , \xi_{j m-1} \end{displaymath}$

are adjoint vectors obtained by solving the linear systems of equations,

$\begin{eqnarray*} (A - \mu E ) \xi_{j 1} &=& \xi_1 \ (A - \mu E ) \xi_{j 2} &=... ... \vdots && \vdots \ (A - \mu E ) \xi_{j m-1} &=& \xi_{j m-2} \end{eqnarray*}$

For a complex eigenvalue

$\alpha + i \beta$

of multiplicity m having only one eigenvector

$\begin{displaymath} \xi_j = a_j + i b_j \end{displaymath}$

we have the following Jordan block of order 2m,

$\begin{displaymath} J_{2m} (\alpha , \beta)= \left( \begin{array}{cccccc} \alp... ... 0 & 0 & \dots &\dots & -\beta & \alpha \end{array}\right). \end{displaymath}$

On the diagonal we have square blocks

$\begin{displaymath} \left( \begin{array}{cc} \alpha & \beta\ -\beta & \alpha \end{array}\right) \end{displaymath}$

Above the diagonal we have identity matrices

$\begin{displaymath} \left( \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right) \end{displaymath}$

All other elements in

$J_{2m} (\alpha , \beta)$

are zeros. The corresponding set of vectors in C looks as

$\begin{displaymath} a_j\; b_j\; a_{j 1}\; b_{j 1} \; \dots \; a_{j m-1}\; b_{j m-1} \end{displaymath}$

where

$a_{j k}$

and

$b_{j k}$

are real and imaginary parts of the adjoint vector

$\xi_{j k}.$

The canonical form of a linear operator

$\begin{displaymath} A: \; {\rm R}^n \; \to \; {\rm R}^n \end{displaymath}$

has the following structure

$\begin{displaymath} \bar A = \left( \begin{array}{cccccccc} \vdots & \dots & \d... ... \dots &\dots &\dots &\ddots & \ddots \ \end{array} \right) \end{displaymath}$

Next:ExercisesUp:Canonical forms of linearPrevious:Linear mapping of the plane

Sergey Nikitin 2004-10-22