1. We are given a matrix $$A=\begin{bmatrix}1&2&0&-1\\3&0&1&2\\2&-1&3&1\\1&4&2&0\end{bmatrix} \in \mathbb{R}^{4 \times 4}$$ and some properties about it and other matrices. 2. First, note the property $$AA^* = A^*A$$ means $$A$$ is a normal matrix (where $$A^*$$ is the conjugate transpose). 3. Because $$A$$ is normal, there exists a unitary matrix $$U$$ such that $$U^* A U = D$$ where $$D$$ is diagonal with eigenvalues of $$A$$. 4. Also, $$U^*U = UU^* = I$$ indicates $$U$$ is unitary. 5. The inequality $$\det(A) \le \prod_{i=1}^n a_{ii}$$ refers to the determinant being less than or equal to the product of diagonal elements when $$A$$ is positive semidefinite or similar. 6. For matrices $$A$$ and $$B$$, the inequality $$\text{trace}(AB) \le \text{trace}(A) \cdot \lambda_{\max}(B)$$ uses the maximum eigenvalue $$\lambda_{\max}(B)$$ and the fact that traces and eigenvalues relate in this way under certain assumptions. 7. $$R_A(x) = \frac{x^T A x}{x^T x}$$ is the Rayleigh quotient of $$A$$ with respect to vector $$x$$. 8. The Rayleigh quotient is bounded by the smallest and largest eigenvalues of $$A$$: $$\lambda_{\min}(A) \le R_A(x) \le \lambda_{\max}(A)$$. 9. The 3 column vectors in $$\mathbb{R}^2$$ given: $$\begin{bmatrix}1 \\ 1\end{bmatrix}, \begin{bmatrix}2 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1\end{bmatrix}$$ are points/vectors for geometric reference. This summary explains the matrix properties and inequalities related to eigenvalues, unitary diagonalization, and Rayleigh quotients.