a.caregnato
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Is [math]T^{-1}[/math] not the matrix inverse of [math]T[/math]?
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.
Thank you for your answer.
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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:
[math] \dot x = \textbf{Fx} + \textbf{G}u [/math]
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:
[math] A = \left| \begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
a & b & c \\ \end{array} \right|.[/math]
For that, the book shows a Linear Transformation in the variable x:
[math] \textbf{x} = \textbf{Tz} [/math]
Which leads to (see first equation):
[math] \dot z =T^{-1} \textbf{FTz} + T^{-1}\textbf{G}u [/math]
The equation for A is:
[math]\textbf{A} = T^{-1} \textbf{FT} [/math]
Where T-1is defined as:
[math] T^{-1} = \left| \begin{array}{ccc}
t1 \\
t2 \\
t3 \\ \end{array} \right|.[/math]
Writing everything in therms of T-1:
[math]\textbf{A} T^{-1} = T^{-1} \textbf{F} [/math]
Now, the problem:
[math]\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|
[/math]
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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Matrix multiplication and Linear Transformation
in Linear Algebra and Group Theory
Posted · Edited by a.caregnato
You're right, I've got it now.
[math]t1,t2,t3[/math] are row vectors (1x3), thats the only way We'll have a matrix T-1 with 3x3 dimensions.
Thank you for your help, ajb.