Transition matrix problem

Aug 2016
38
2
Romania
I have the following problem that asks me what is the transition matrix from B to B'.In the second image you can view the answer but I don't understand how that answer appeared.
Can someone explain me the steps to solve this problem?
I am also confused because the answer is a 3x3 matrix and all the matrices are in the form of a 2x2 which is not the same.

https://ibb.co/YhSQ3Q1
https://ibb.co/y6DSrQf
 

romsek

Math Team
Sep 2015
2,773
1,552
USA
$A$ is the solution to

$B' = A^T B$

for example

$\begin{pmatrix}1&0\\0&2\end{pmatrix} =
\dfrac 1 2 \begin{pmatrix}2&0\\0&0\end{pmatrix}+
0 \begin{pmatrix}0&3\\-3&0\end{pmatrix}+
\dfrac 2 5\begin{pmatrix}0&0\\0&5\end{pmatrix}$
 
Aug 2016
38
2
Romania
$A$ is the solution to

$B' = A^T B$

for example

$\begin{pmatrix}1&0\\0&2\end{pmatrix} =
\dfrac 1 2 \begin{pmatrix}2&0\\0&0\end{pmatrix}+
0 \begin{pmatrix}0&3\\-3&0\end{pmatrix}+
\dfrac 2 5\begin{pmatrix}0&0\\0&5\end{pmatrix}$
It will be something like this?
https://ibb.co/vxVtVCj

Also in your example A^(T) means a transposed matrix?