i have a question regarding the image of a quadratic operator.
Suppose I have $A\in\Re^{5\times 5}$ a symmetric matrix whose $a_{ij}$ entries are strictly positive and I am interested in the domain of the function $f:\Re^n\mapsto\Re$ define as the quadratic form $$f(X)=X^TAX$$ where $X=[X_1;X_2]$ with $X_1\in \Re^{+}\times\Re^{+}\times\Re^{+}$ and $X_2\in \Re^{-}\times\Re^{-}$.
It is intuitive that $f(X)$ takes values in $(0,\infty)$ but is there a way to prove it or some results I can use? Thanks
It suffices to consider the case $n=2$ and $f:X\in\mathbb{R}^+\times\mathbb{R}^-\rightarrow X^TAX$.
If $A=\begin{pmatrix}1&1\\1&1\end{pmatrix}$, then $A\geq 0$ and $im(f)=\mathbb{R}^+$.
If $A=\begin{pmatrix}1&2\\2&1\end{pmatrix}$, then $im(f)=\mathbb{R}$ (take $X=[1,0]^T$ and $X=[1,-1]^T$).