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!