Factoring a quadratic function

Question

Suppose you have a vector of variables $x\in\mathbb{R}^n$ and a function defined by $h(x)=\sum_{i=1}^n\sum_{j=1}^n c_{ij}x_ix_j$, where $c_{ij}$ are rational numbers. My questions are, is it possible to write $h$ as the product of two linear functions $f(x)=\sum_{i=1}^n \pi_ix_i$ and $g(x)=\sum_{i=1}^n \alpha_ix_i$, such that $h(x)=f(x)g(x)$? And if so, how do you renover the coefficients $\pi_i$ and $\alpha_i$. Thanks in advance.

Answer 1

Let $M$ be the matrix of your quadratic form $h$ : $M=(c_{i,j})_{1\le i,j\le n}$, such that if $X=\begin{pmatrix}x_1 \\ \vdots \\ x_n\end{pmatrix}$, $$h(x)={}^tXMX$$ What you are asking can be translated in : $M$ is of rank $1$. So all the columns have to be proportional. If one of these columns, say $C_1$, is non null, then all the others have to be of the form $C_j=\lambda_j C_1$.

Now this means $M={}^t\Lambda \Gamma$, where $\Lambda=\begin{pmatrix}\lambda_1 \\ \vdots \\ \lambda_n\end{pmatrix}$ and $\Gamma=\begin{pmatrix}\gamma_1 \\ \vdots \\ \gamma_n\end{pmatrix}$, you can choose $$f(X)={}^t\Lambda X\text{ and }g(X)={}^t\Gamma X$$

Answer 2

No, you can't represent $n^2$ numbers from $2n$ numbers (unless $n\le2 $.)