Turn one vector into another vector

[askmathquestions]

This might seem like a novice question, but let's suppose we have a vector $x$ and we want to turn it into vector $y$. Well, what square matrix multiplied on $x$ accomplishes this?

As an example, let's work with a $2 \times2$ case:

$x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ and $y = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$, then what transform $T = \begin{bmatrix} t_{1,1} & t_{1,2} \\ t_{2,1} & t_{2,2} \end{bmatrix}$ makes the equation $Tx = y$ true?

 

[topsquark]

This might seem like a novice question, but let's suppose we have a vector $x$ and we want to turn it into vector $y$. Well, what square matrix multiplied on $x$ accomplishes this?

As an example, let's work with a $2 \times2$ case:

$x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ and $y = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$, then what transform $T = \begin{bmatrix} t_{1,1} & t_{1,2} \\ t_{2,1} & t_{2,2} \end{bmatrix}$ makes the equation $Tx = y$ true?

Well,
$\begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} t_{11} & t_{12} \\ t_{21} & t_{22} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$

So
$y_1 = t_{11} x_1 + t_{12} x_2$

$y_2 = t_{21} x_2 + t_{22} x_2$

You have two equations in four unknowns so you can set some conditions on T if you like. But that's the proceedure.

-Dan

 

[askmathquestions]

This is where I got stuck, because how do we solve for the $t_{i,j}$ components? Or, what conventional restrictions are there?

 

[topsquark]

This is where I got stuck, because how do we solve for the $t_{i,j}$ components? Or, what conventional restrictions are there?

You have two equations in four unknowns. So you can solve for two of the in terms of the remaining two unknowns. So you can set two more conditions.

There are a number of restrictions you can impose. You can set T to be unitary, det(T) = 1, Tr(T) = 0, you can make T symmetric or antisymmetric, skew-symmetric, etc. Or you could just simply say, "I want $t_{11} = \pi$ and $t_{21} = 0$." As long as you don't run into a contradiction you can do just about anything you like. Of course you would want to tailor your conditions to whatever problem you are working with.

-Dan

 

[PeroK]

One approach would be to express the vectors in polar form:
$$\vec v_1 = (x_1, x_2) = (r\cos \theta, r\sin \theta)$$$$\vec v_2 = (y_1, y_2) = (R\cos \phi, R\sin \phi)$$Then the rotation matrix for the angle $\phi - \theta$, followed by the identity matrix multiplied by $\frac R r$ would do the trick.