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Let $$ A= \begin{pmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 1\\ \end {pmatrix} $$

Find a nonsingular matrix $P$ such that $P^{-1}AP$ is in Jordan form.

The course I am taking uses the textbook "Matrices and Linear Transformation" by Cullen.

The example in the book explains how to find $P$ if I know the characteristic polynomial of $A$.

When I tried to find the characteristic polynomial of this matrix, I got TWO eigenvalues: 0 and 1.

According to the example, I need to first find the matrix J which A is similar to.

Theorem 5.12 in my textbook states:If $A \in F_{n\times n}$ has characteristic polynomial $c(x)=\det(xI-A)=\prod^{r}_{i=1}(x-\lambda_{i})^{s_{i}}$ then $A$ is similar to a matrix $J$ with the $\lambda_{i}$ on the diagonal, zeros and ones on the subdiagonal, and zeros elsewhere.

Am I correct in saying $$J= \begin{pmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\\ \end{pmatrix} $$

Please help.

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1 Answer 1

up vote 1 down vote accepted

Yes, you're correct. The Jordan blocks in your case are $$J_2(0) = \pmatrix{0 & 0\\ 1 & 0} \\ J_1(1) = \pmatrix{1}.$$ You need to look at Theorem 5.13 in your book.

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J.D I am trying to follow an example in the book which uses the similar Jordan matrix to then determine the matrix P. However, the example works with a matrix A where there was only one eigenvalue. The example is on page 204, can you tell me how I would have to modify this so that it works for this matrix? –  sarah jamal Jul 15 '12 at 18:09
can you help me find P? the example in the book works with a matrix which has one eigenvalue. The example is on pg204. –  sarah jamal Jul 15 '12 at 18:22
@sarah: Google Books doesn't show me p. 204 in the book. –  joriki Jul 16 '12 at 11:10
I guess this is now covered by copper.hat's answer on your other question: click here. –  user2468 Jul 16 '12 at 13:32

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