Jordan Canonical Form transition matrix

Question

I have this matrix $M$

$M = \begin{bmatrix} 1 & 1 & 1\\ 2 & 1 & -1\\ 0 & -1 & 1 \end{bmatrix}$

And I was asked to put it into Jordan Canonical Form.

I did this, and obtained

if $M$ = $SQS^{-1}$

Then $S = \begin{bmatrix} -3 & 0 & -1\\ 4 & -1 & -1\\ 2 & 1 & 0 \end{bmatrix}$

And $Q = \begin{bmatrix} -1 & 0 & 0\\ 0 & 2 & 1\\ 0 & 0 & 2 \end{bmatrix}$ is the matrix of jordan blocks

I am now asked to "find the appropriate transition matrix to the basis in which the the original matrix assumes its Jordan form." Needless to say I have no idea what it is asking and any clarification would be largely appreciated.

Gracias

Answer 1

You've already answered the question. $Q$ is the "Jordan canonical form", and $S^{-1}$ is "the appropriate transition matrix to the basis in which the original matrix assumes its normal form."