Matrix multiplication and Linear Transformation

[a.caregnato]

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!

[ajb]

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.

[a.caregnato]

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.

Edited December 24, 2015 by a.caregnato

[ajb]

[math]T[/math] 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.

[a.caregnato]

[math]T[/math] 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.

 

[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.

Edited December 24, 2015 by a.caregnato

[ajb]

[math]t1,t2,t3[/math] are row vectors (1x3), thats the only way We'll have a matrix [math]T^-1[/math] 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.