Matrix multiplication and Linear Transformation

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Hello everybody.

I'm having a little bit of trouble understanding a passage of my textbook regarding a linear transformation and matrix multiplication, I wonder if you could help me out.

So, I have this equation:

$\dot x = \textbf{Fx} + \textbf{G}u$

Where F is some 3x3 matrix and x a 3x1 array. For now, these are the important variables. So, my objective is putting F in a specific format called control canonical form (A), which is:

$A = \left| \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \\ \end{array} \right|.$

For that, the book shows a Linear Transformation in the variable x:

$\textbf{x} = \textbf{Tz}$

Which leads to (see first equation):

$\dot z =T^{-1} \textbf{FTz} + T^{-1}\textbf{G}u$

The equation for A is:

$\textbf{A} = T^{-1} \textbf{FT}$

Where T-1is defined as:

$T^{-1} = \left| \begin{array}{ccc} t1 \\ t2 \\ t3 \\ \end{array} \right|.$

Writing everything in therms of T-1:

$\textbf{A} T^{-1} = T^{-1} \textbf{F}$

Now, the problem:

$\left| \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \\ \end{array} \right| \left| \begin{array}{ccc} t1 \\ t2 \\ t3 \\ \end{array} \right| = \left| \begin{array}{ccc} t1 \textbf{F} \\ t2 \textbf{F} \\ t3 \textbf{F} \\ \end{array} \right|$

I don't understant the right part of the equation. How can I multiply T-1, which is a 3x1 array, with the 3x3 F matrix? Why the book shows a array with every single term of T-1 multiplying F? I apologize if this is some stupid question but linear algebra isn't my strong suit.

Thanks!

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ajb    1567

Is $T^{-1}$ not the matrix inverse of $T$?

It looks like you are trying to show that A and F are similar.

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Is $T^{-1}$ not the matrix inverse of $T$?

It looks like you are trying to show that A and F are similar.

Yes, is the inverse. I'm trying to find the terms of the inverse transformation matrix (t1,t2,t3) which will "turn" F into A (It's easy to figure out T knowing T-1) . But that last equation doens't make any sense to me.

Edited by a.caregnato

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ajb    1567

$T$ is an nxn matrix and so its inverse is also an nxn matrix. I do not understand what you have written, but this could be a notational issue.

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$T$ is an nxn matrix and so its inverse is also an nxn matrix. I do not understand what you have written, but this could be a notational issue.

You're right, I've got it now.

$t1,t2,t3$ are row vectors (1x3), thats the only way We'll have a matrix T-1 with 3x3 dimensions.

Thank you for your help, ajb.

Edited by a.caregnato

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ajb    1567

$t1,t2,t3$ are row vectors (1x3), thats the only way We'll have a matrix $T-1$ with 3x3 dimensions.

That was what I was wondering. Then notation is not great in my opinion.

Thank you for your help, ajb.

No problem.