Similarity Transformation misunderstanding

Question

All of my scripts and textbooks about control theory introduce the similarity transformation. It is needed to work in the state space. However, none of them have any examples. I don't get what they are doing.

Then this exercise came without solution, which perfectly shows that I don't get it:

By means of a similarity transoformation P can be transformed into $\tilde{P}=\begin{bmatrix}\tilde{P}_1 \\ 0\end{bmatrix}.$Determine the necessary state transformation matrix T.

So a similarity transformation is $\tilde{P}=TP$. I search for T. My first problem is that I don't see what $\tilde{P}_1$ is. In my view P is a matrix of undefined size. Therefore the 1 is an index. But I'd need two indexes for a matrix element. So my assumption is probably wrong.

My second problem is how to get T. If I know what $\tilde{P}_1$ is I could do something like $T=\tilde{P}_1P^-1$. But then I also need to know what P is or how else am I supposed to invert it?

This exercise should be rather simple since it gave only 1 point. I hope it fits the requirements for Mathematics StackExchange and is not too trivial. I am glad for any advice which points me in the right direction of similarity transformations in general.

EDIT: Apparently some matrix P is needed. There is never mentioned that it is this P, but in a previous exercise P is mentioned as:

The controllability matrix $P=\left[\begin{array}{lll}{A^{2} B} & {A B} & {B}\end{array}\right]$ of a state-space model $(A,B,C,D)$ has the following SVD. Determine the kernel and image space of $P$. $$ P=\left[\begin{array}{lll}{u_{1}} & {u_{2}} & {u_{3}}\end{array}\right]\left[\begin{array}{ccc}{\sigma_{1}} & {0} & {0} \\ {0} & {\sigma_{2}} & {0} \\ {0} & {0} & {\sigma_{3}}\end{array}\right]\left[\begin{array}{c}{v_{1}^{T}} \\ {v_{2}^{T}} \\ {v_{3}^{T}}\end{array}\right] $$

EDIT 2: Since there seems that some crucial information is missing I will add the first exercises as well. I don't think it contains information that will help, but I'll give it a try. I strongly assumed those exercises are independent. What's following is exercises 1.a) so it actually came first. The first edit was 1.b) and what I actually asked is in 1. c)

Each element of the following transfer function G(s) has the bode diagram shown in the succeeding bode plots. Estimate $\|G(s)\|_{\infty} $ $$ G(s)=\left[\begin{array}{cc}{\frac{2}{s^{2}+0.2 s+1}} & {0} \\ {0} & {\frac{5}{s^{2}+0.4 s+4}}\end{array}\right] $$

Answer 1

A similarity transformation for a state space model $(A,B,C,D)$ can be described with

\begin{align} \hat{A} &= T\,A\,T^{-1}, \\ \hat{B} &= T\,B, \\ \hat{C} &= C\,T^{-1}, \\ \hat{D} &= D. \end{align}

So the transformed controllability matrix would be

\begin{align} \tilde{P} &= \begin{bmatrix} \hat{A}^2 \hat{B} & \hat{A}\,\hat{B} & \hat{B} \end{bmatrix} \\ &= T \begin{bmatrix} A^2 B & A\,B & B \end{bmatrix} \\ &= T\,P. \end{align}

It can be noted that $T$ should always be invertible (otherwise $\hat{A}$ and $\hat{C}$ can not be calculated). From this it also follows that the rank of $P$ should equal the rank of $\tilde{P}$. Now since $\tilde{P}$ should contain a row of zeros means that it can't be full rank so neither can $P$. This would mean that either one or more of the three $\sigma$ should be zero or $u_i\in\mathbb{R}^n$ with $n>3$. However in the previous exercise you had to calculate the kernel of $P$ so I assume that one of these requirements would be satisfied. From this you can also deduce that $T$ could be obtained using

$$ T = \begin{bmatrix} T_1 \\ T_2 \end{bmatrix}, $$

with $T_2$ the left null space of $P$ and $T_2$ such that $T$ is square and full rank.