I know the basis about Schur-Complement. Anyway, while looking at this inequality to apply it in order to solve for $\lambda$ such that the the matrix is definite positive, I got a little bit confused because the lack of non inverse terms.
${ X }^{ T }\left( PA+{ A }^{ T }P \right) X<-\lambda { X }^{ T }AX$
May I... ?
$\begin{bmatrix} 0 & { X } \\ { X }^{ T } & { \left( PA+{ A }^{ T }P+\lambda A \right) }^{ -1 } \end{bmatrix}$
Or,
$\begin{bmatrix} { X }^{ T }\left( PA+{ A }^{ T }P+\lambda A \right) X & { P } \\ P & 0 \end{bmatrix}$
Or even,
$\begin{bmatrix} { X }^{ T }\left( PA+{ A }^{ T }P \right) X & X \\ { X }^{ T } & -\lambda { A }^{ -1 } \end{bmatrix}$
I'm sorry if this is a silly or even stupid question, but i'm lost :( Any references, books, or examples that You could recommend me?
I don't see the point of using Schur's complements here.
In fact, looking for $\lambda$ satisfying inequality
$$X^T\left( PA+A^TP \right) X<-\lambda X^TAX\tag{1}$$
is the same as looking for the bounds of
$$\dfrac{X^TBX}{X^TAX} \ \ \text{where} \ \ B:=PA+A^TP \tag{2}$$
which is a classical issue :
If $A$ is positive-definite, consider a Cholesky decomposition $A=C^TC$, set $Y=CX$, transforming (2) into the Rayleigh quotient :
$$\dfrac{YC^{-T}BC^{-1}Y}{Y^TY} $$
which is known to take all values in interval $[\lambda_{min},\lambda_{max}]$ where $\lambda_{min},\lambda_{max}$ are the extreme eigenvalues of matrix $D:=C^{-T}BC^{-1}$.
Connected :
Proof: Ratio of matrix traces and difference of traces
Ratio of two quadratic vector forms