Why the largest eigenvalue is the bound? I am new to Linear Algebra, say introduction level (know the definition and some basic properties, theories about it but not really familiar with doing math on it). Please explain to me:
If A is an inverse of a positive definite matrix, then why all of its eigenvalues are positive and why $\frac{x^TAAx}{1+x^TAx}$ is bounded above by the largest eigenvalue of A?
Can you recommend some books that offer a higher level than introduction that can help me get over these kind of problems? I'm learning Machine Learning by the way.
Thank you.
 A: $A$ is inverse of positive definite matrix then $A$ is positive definite. Positive definite matrices have real parts of eigenvalue positive ( in case of symmetric matrices, where eigenvalues are real, then simply eigenvalues are positive).
Since you mention, eigenvalues are real so I assume, you're talking about symmetric matrices. This is relatively easy to analyze case since, in this case eigenvectors are orthogonal. 
If $q_{i}$'s are eigenvectors (hence $Aq_{i}=\lambda_{i}q_{i}$, then   any $x$ can be written as x=$\Sigma\alpha_{i}q_{i}$, and it follows that
$\frac{x^TAAx}{1+x^TAx} = \frac{\Sigma\alpha_{i}\lambda^{2}_{i}}{1+\Sigma\alpha_{i}\lambda_{i}}$
It is now easy to see that maximum would occur when all $\alpha_{i}$ are zero except one that corresponds to largest eigenvalue. Hence $\frac{\lambda^{2}_{max}}{1+\lambda_{max}}\leq \lambda_{max}$. Thus the quantity is bounded by largest eigenvalue. 
In order to be able to do such manipulation with eigenvalues and eigenvectors, I suggest you read about 


*

*Chapter on eigenvalue computation from Numerical Linear Algebra by (Trefethen and Bau) this book used to be available online

*About Rayleigh quotient 

*For much more deeper theoretical understanding, you can read about Min-Max theorems about eigenvalue 

*It will be also useful to read about singular values and singular vector for more general cases where eigenvalues are not positive or real 

