How can you manipulate a vector to create different matrices?

[Jhenrique]

Given a vector r = (x, y, z) is possible to make some manipulation for get the matrix:
$$\begin{bmatrix} 0 & z & -y\\ -z & 0 & x\\ y & -x & 0\\ \end{bmatrix}$$
and this matrix too:
$$\begin{bmatrix} x & 0 & 0\\ 0 & y & 0\\ 0 & 0 & z\\ \end{bmatrix}$$
?

 

[Office_Shredder]

What do you mean by "manipulation"? You just wrote those matrices down, you now have them to do whatever you want with them. Are you asking whether the map
$$(x,y,z) \mapsto \left( \begin{array}{ccc} 0 & z & -y\\ -z & 0 & x\\ y & -x & 0 \end{array} \right)$$
is linear or something to that effect?

 

[Jhenrique]

manipulation in the sense of add, subtract, multiply, divide... algebraic/matrix manipulation

 

[Jhenrique]

I discovered how make the 1nd transformation!

Let [r] the notation for the first matrix of my post #1, it's is given by: $[\vec{r}] = \sqrt{\vec{r}\otimes \vec{r}-r^2 I}$

However, I still don't know how get the second matrix...

 

[AlephZero]

A hint:

$\begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x \end{pmatrix}$

Then, use Kronecker products...

 

[Jhenrique]

A hint:

$\begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x \end{pmatrix}$

Then, use Kronecker products...

Give me an example