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Matrix multiplication and Linear Transformation


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#1 a.caregnato

a.caregnato

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Posted 24 December 2015 - 01:45 PM

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 is some 3x3 matrix and 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-1 is 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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#2 ajb

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Posted 24 December 2015 - 02:00 PM

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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#3 a.caregnato

a.caregnato

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Posted 24 December 2015 - 02:08 PM

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

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

Yes, bbf1f4d512921e33d475932972c88f9c-1.png is the inverse. I'm trying to find the terms of the inverse transformation matrix (t1,t2,t3) which will "turn" into A (It's easy to figure out T knowing T-1) . But that last equation doens't make any sense to me.

 

Thank you for your answer.


Edited by a.caregnato, 24 December 2015 - 02:09 PM.

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#4 ajb

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Posted 24 December 2015 - 02:11 PM

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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#5 a.caregnato

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Posted 24 December 2015 - 02:57 PM



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, 24 December 2015 - 02:57 PM.

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#6 ajb

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Posted 24 December 2015 - 02:59 PM

t1,t2,t3 are row vectors (1x3), thats the only way We'll have a matrix T<sup>-1</sup> 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.
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