Problem Analysis #1

For a square matrix $A$, the matrix exponential $e^{A}$ is defined by$e^{A} = E + \sum_{k=1}^{\infty} \frac{1}{k!}A^{k},$where $E$ is the identity matrix.

• [(1)] Let $A= \begin{pmatrix} -1 & 4 & 4 \\ -5 & 8 & 10 \\ 3 & -3 & -5 \end{pmatrix}.$ Compute the matrix exponential $e^{A}$.
• [(2)] Consider the matrices $A_x= \begin{pmatrix} 0 & -x & 0 \\ x & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix},\qquad B= \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix},$ where $x$ is a real number, and define the vector $a= \begin{pmatrix} 1 \\[2mm] 1 \\[2mm] e \end{pmatrix}.$ Let $C = B e^{A_x}$. Determine the minimum and maximum values of $\frac{y^{T} C y}{y^{T}y}$ over all nonzero real vectors $y\in\mathbb{R}^{3}$.
• [(3)] Let $f(z)=\frac{1}{2}z^{T} C z - a^{T}z, \qquad z= \begin{pmatrix} z_{1}\\ z_{2}\\ z_{3} \end{pmatrix},$ where $C$ is the matrix from part (2). Find the value of $\sqrt{z_1^2+z_2^2+z_3^2}$ for the vector $z$ satisfying $\frac{\partial f}{\partial z_{1}} = \frac{\partial f}{\partial z_{2}} = \frac{\partial f}{\partial z_{3}} =0.$ Let $\lambda_1,\lambda_2,\lambda_3$ be the eigenvalues of the matrix $A$ in part {\rm (1)}. Let $m$ and $M$ be the minimum and maximum values, respectively, obtained in part {\rm (2)}, and let$R = \sqrt{z_1^2+z_2^2+z_3^2}$be the value obtained in part {\rm (3)}. Compute$\lambda_1\lambda_2\lambda_3 \cdot m \cdot R^2 \cdot \bigl(\lambda_1^2+\lambda_2^2+\lambda_3^2\bigr).$\