Matrix Manipulation/Transformation

I have three $n \times 1$ matrices:

$$[A] = \left[ \begin{array}{c} a_1 \\ a_2 \\ \vdots\\ a_n \end{array} \right]$$

$$[B] = \left[ \begin{array}{c} b_1 \\ b_2 \\ \vdots\\ b_n \end{array} \right]$$

$$[C] = \left[ \begin{array}{c} a_1 b_1 \\ a_2 b_2 \\ \vdots\\ a_n b_n \end{array} \right]$$

I want $[C]$ as a function of $[A]$ and $[B]$. To put it another way, if,

$$[C]_{n \times 1} = [F]_{n \times n} [B]_{n \times 1}$$

$$[F]_{n \times n} = \left[ \begin{array}{ccccc} a_1 &0&\dots&0&0 \\ 0&a_2&0&\dots&0 \\ \vdots&&\ddots&&\vdots\\ 0&\dots&0&a_{n-1}&0\\ 0&0&\dots&0&a_n \end{array} \right]$$ then, what is $[F]$ as a function of $[A]$?

EDIT

If what I am asking cannot be done, just tell me if there is a name for transformation $\Gamma$, below:

$$\Gamma([A]) = [F]$$

Or should I just define it myself in the manuscript.

• Is it possible what you're asking cannot be done? How did you come across this question? – Mike Earnest Feb 3 '15 at 23:43
• Do you mean to ask, what is another matrix $G$ such that $C = GA$? If so, then I think you have your answer in front of you. – Mnifldz Feb 3 '15 at 23:44
• I'm not sure if it is possible. I'm writing a Finite difference library in C++ and I need that for the theoretical documentation. – Eliad Feb 4 '15 at 0:11
• @Mnifldz, No I do not mean that. – Eliad Feb 4 '15 at 0:20
• Sometimes the notation diag$(a_1,\dots,a_n)$ is used to mean the matrix $F$ you want. – Mike Earnest Feb 4 '15 at 1:39