Compute the Jordan canonical form of a given matrix

Compute the Jordan canonical form of A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -2 \\ 0 & 1 & 3 \end{bmatrix}.


We begin by computing the Smith normal form of xI-A. Evidently the following sequence of ERCOs achieves this.

  1. R_2 + xR_3 \rightarrow R_2
  2. C_3 + (x-3)C_2 \rightarrow C_3
  3. -C_2 \rightarrow C_2
  4. R_2 \leftrightarrow R_3
  5. R_1 \leftrightarrow R_2
  6. C_1 \leftrightarrow C_2

The resulting matrix is \begin{bmatrix} 1 & 0 & 0 \\ 0 & x-1 & 0 \\ 0 & 0 & (x-1)(x-2) \end{bmatrix}. So the invariant factors of A are x-1 and (x-1)(x-2), and the elementary divisors are thus x-1, x-1, and x-2. So the Jordan canonical form of A is \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}.

Advertisements
Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: