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Suppose $A$ is a linear transformation of a 3-dim vector space $V$, defined as $$A(\epsilon_1,\epsilon_2,\epsilon_3)=(\epsilon_1,\epsilon_2,\epsilon_3) \begin{pmatrix} -10 & 12 & 7\\ -3 & 4 & 2\\ -13 & 15 & 9 \end{pmatrix}, $$here $\{\epsilon_i\}$ is a basis of $V$.

Is there a concise way to find the transition matrix to a new basis under which the linear operator $A$ has a matrix of Jordan form? And what's behind the solution?

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Does this help? – user21436 Apr 22 '12 at 6:42
Have you worked out the eigen values, firstly, so that you can work with transformations that have eigen value $0$? – user21436 Apr 22 '12 at 6:50
I assume you're working over complex field, and can you tell us what the characteristic polynomial of this transformation? – user21436 Apr 22 '12 at 6:52
So, tell us what the characteristic polynomial and roots are? – user21436 Apr 22 '12 at 6:55
The characteristic polynomial is $\lambda^3-3\lambda^2+3\lambda-1$, and the roots are all equal to 1. – rhenskyyy Apr 22 '12 at 6:59
up vote 2 down vote accepted

Here are a couple of observations:

  1. The characteristic polynomial is $\lambda^3-3\lambda^2+3\lambda-1=0$ and therefore the only eigen value is $1$.

  2. We observe that $A-I$ has rank $2$ and hence $\dim(\ker(A-I))=1$ from the Rank-Nullity theorem.

  3. We know that the Jordan form for the matrix is $$J=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1 \end{bmatrix}$$

  4. We are interested in finding a matrix $P$ such that $$P^{-1}AP=J=I+\begin{bmatrix} 0&1&0\\0&0&1\\0&0&0\end{bmatrix}$$

Setting $P=\begin{bmatrix} \biggl |& \biggl|&\biggl|\\x_1&x_2 &x_3\\\biggl|&\biggl|&\biggl|\end{bmatrix}$, and observing that, $AP=PJ$, can you solve the resulting system of vectors?

A couple of preliminaries:

  • Verify that $AP=\begin{bmatrix}\biggl| & \biggl| &\biggl|\\Ax_1& Ax_2 &Ax_3\\ \biggl| &\biggl| & \biggl|\end{bmatrix}$

  • Verify that $PJ=\begin{bmatrix}\biggl| & \biggl| &\biggl|\\x_1& x_1+x_2 &x_2+x_3\\ \biggl| &\biggl| & \biggl|\end{bmatrix}$

The steps 1 and 2 were already known to the OP. I'll post more hints as I work out.

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