How do I find matrix that satisfies following conditions?

Question

How do I find matrix $A$ with integer entries given two $2\times 1 $ vectors $\vec{x}, \vec{a} $ such that

$$\vec{x} = A \vec{a}$$

Answer 1

Let $A = \left(\begin{array}{cc} a & b \\ c & d\end{array}\right)$. Then we want to solve

$$\left(\begin{array}{c} x_1 \\ x_2 \end{array}\right) = \left(\begin{array}{cc} a & b \\ c & d\end{array}\right)\left(\begin{array}{c} a_1 \\ a_2 \end{array}\right).$$

We can write this as a system of two equations:

$$a a_1 + ba_2 = x_1 \\ ca_1+da_2 = x_2$$

Or to put it in a more tangible form..

$$a_1a + a_2b + 0c + 0d = x_1 \\ 0a + 0b + a_1c + a_2 d = x_2$$

This is a very underdetermined system. You have four variables and two equations so there is no unique solution to this. To get a unique solution you have to place some constraints on the system. A natural one is to ask maybe that $A$ be symmetric (or more generally self-adjoint) but this only reduces the number of variables to $3$ so you still have an underdetermined system. If you also ask that $A$ be orthogonal (or more generally unitary) then you might get a unique solution if $\lVert \vec{x}\rVert = \lVert \vec{a}\rVert$.