Jump to content

Matrix multiplication and Linear Transformation


Recommended Posts

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!

Link to comment
Share on other sites

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, bbf1f4d512921e33d475932972c88f9c-1.png 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 by a.caregnato
Link to comment
Share on other sites

[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 by a.caregnato
Link to comment
Share on other sites

[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.

Link to comment
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now
×
×
  • Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.