<s>[INST] General form of multiplication of a matrix by its transpose When multiplying $A^T*A$, does it usually end up in the form $\begin{pmatrix} a&b\\b&-2b\end{pmatrix}$? If not, are there any special cases where it does? If so, why? 
 [/INST] Per the comment, we generally have
$$
\begin{pmatrix} a&b\\c&d\end{pmatrix} \begin{pmatrix} a&c\\b&d\end{pmatrix} = \begin{pmatrix} a^2+b^2 & ac+bd \\ ac+bd & c^2+d^2\end{pmatrix} \overset{?} = \pmatrix{\alpha & -\frac 12 \beta\\ - \frac 12 \beta  & \beta} 
$$
Your question amounts to asking when $a,b,c,d$ are such that
$$
ac + bd = - \frac 12 (c^2 + d^2)
$$
Given any $c,d$, there is a line of solutions $(a,b)$ to this equation.
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