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