In my adaptive control textbook that I am using, we are using the following: $$\left\|\Phi^T\left(\mathbf{x}\right)\Phi\left(\mathbf{x}\right)\right\| \geq \left\|\Phi\left(\mathbf{x}\right)\Phi^T\left(\mathbf{x}\right)\right\|$$
where $\Phi\left(\mathbf{x}\right) \in \mathbb{R}^m$ and are "implicit" functions of time. Then, he claims that the following is true: $$\alpha_0 I \leq \int_{t}^{t+T}\Phi\left(\mathbf{x}\right)\Phi^T\left(\mathbf{x}\right)d\tau \leq \left(\int_{t}^{t+T}\Phi^T\left(\mathbf{x}\right)\Phi\left(\mathbf{x}\right)d\tau\right)I = \beta_0 I$$ assuming that the integrals are finite. Finally, given the second inequality, we see that
$$\alpha_0 \leq \left\|\Phi\left(\mathbf{x}\right)\Phi^T\left(\mathbf{x}\right)\right\|$$
Where $\alpha_0$ is the infimum of the eigenvalues or less than those of $\Phi\left(\mathbf{x}\right)\Phi^T\left(\mathbf{x}\right)$, which is supposed to be positive definite. Can someone explain why the three inequalities are true?
For any real vector $u$ we can show that $\lVert u^T u \rVert = u^T u = \lVert u u^T \rVert_2$.
Note that $uu^Tu=(u^Tu)u$, which means $(u^Tu, u)$ is an eigenpair of $uu^T$. Since it is symmetric and a rank-1 matrix $\lVert u u^T \rVert_2=\rho(uu^T)=u^Tu$ where $\rho(\cdot)$ is the spectral radius.