Lagrange identity on matrices Consider the $k-$dimensional homogeneous linear difference system $ x(n+1) = A(n) x(n) $.
Define $ H(n):= A^T (n) A(n)$.
(a) Prove the Lagrange identity  $$\displaystyle{ || x(n+1) ||_2 ^2 = x^T (n+1) x(n+1) = x^T (n) H(n) x(n) }$$
(b) Show that all eigenvalues of $H(n)$ are real and nonnegative.
(c) Let that all the eigenvalues of $H(n)$ be ordered as $ \lambda_1 (n) \leq \lambda_2 (n) \leq \cdots \leq \lambda_k (n) $. Show that for all $ \displaystyle{x \in \mathbb R ^{ k \times 1} }$, 
$$ \lambda_1 (n) x^T x \leq x^T H(n) x \leq \lambda_k (n) x^T x $$
(d) Using (a)  and (c) show that:
$$ \left( \prod_{i=n_0}^{n-1} \lambda_1 (i) \right ) x^T(n_0) x(n_0) \leq x^T (n)  x(n) \leq  \left( \prod_{i=n_0}^{n-1}\lambda_k (i) \right)x^T(n_0) x(n_0) $$
(e) Show that  $$\displaystyle{\prod_{i=n_0}^{n-1} \lambda_1 (i) \leq || |\Phi( n, n_0) ||^2 \leq \prod_{i=n_0}^{n-1} \lambda_k (i)}$$ 
where $\Phi( n, n_0) = \Phi(n) \Phi^{-1} (n_0)$ and $ \Phi$ is a fundamental matrix of the system.
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I have done (a) and (b) and I need some help for the others.
This is not homework. I am studying for a midterm exam.
Thank's in advance!
P.S This is an exercise from S.N.Elaydi book on difference equations.
edit: I have edit the question (d). I had a typo in one term of the inequality. Now is correct. I am really sorry for that.
Thank you!
 A: Point c) is a basic property of (symmetric) positive definite matrices. Once you know these matrices allow an orthogonal diagonalization $H = P \Lambda P^t$  where $P$ is the (orthogonal) eigenvenctors matrix and $\Lambda$ is diagonal (with positive eigenvalues in its diagonal), so, for any $x$
$$x^T H x = x^T P \Lambda P^T x = y^T \Lambda y = \lambda_1 y_1^2 + \lambda_2 y_2^2  + \cdots\lambda_k y_k^2 $$
where $y=P^t x$, with $|y| = |x|$. 
The above is bounded from below by $\lambda_1 |y|^2=\lambda_1 |x|^2$ and from above by $\lambda_k |y|^2=\lambda_k |x|^2$ 
Hence, in conclusion
$$ \lambda_1 |x|^2 \le x^t H x \le \lambda_k |x|^2$$
For point d): Take for example the upper bound, and apply it, together with a), iteratively:
$$ x_n^T x_n = x_{n-1}^T H_{n-1} x_{n-1}  \le \lambda^{(k)}_{n-1} x_{n-1}^T x_{n-1} =  \lambda^{(k)}_{n-1} x_{n-2}^T H_{n-2} x_{n-2} \le \lambda^{(k)}_{n-1} \lambda^{(k)}_{n-2} x_{n-3}^T x_{n-3}  $$
You shoud get inequalities e) choosing $x(n_0)$ as the vector that attains the supremum in the matrix norm definition, and relating $\Phi$ to $A$ and $H$.
