1. Problem: Given a symmetric matrix $A$ with known eigenvalues and corresponding eigenvectors, find: 1.i. The eigenvector corresponding to a given eigenvalue. 1.ii. The matrix $A$ itself. 2. Problem: Given an approximation to the root of an equation, use the Newton-Raphson method to find a better approximation and determine the root to 4 decimal places. 3. Problem: Given data for a cubic polynomial $f(x)$, use the Newton-Gregory forward difference interpolation formula to determine $f(x)$. 4. Problem: For a linear time variant system described by differential equations: 4.i. Express the system in state-space form. 4.ii. Determine the state transition matrix. 5. Problem: Given functions $u$ and $v$: 5.i. Express in complex form. 5.ii. Show that $u$ and $v$ satisfy the Cauchy-Riemann equations. --- 1.i. Eigenvector corresponding to eigenvalue $\lambda$: - To find eigenvectors for a given eigenvalue $\lambda$, solve the system $$ (A - \lambda I) \mathbf{x} = 0 $$. - This means find non-zero vectors $\mathbf{x}$ in the null space of $A - \lambda I$. 1.ii. Matrix $A$ from eigenvalues and eigenvectors: - Since $A$ is symmetric and diagonalizable, express $A$ as $$ A = PDP^T $$ where $P$ is the orthogonal matrix with eigenvectors as columns and $D$ is the diagonal matrix of eigenvalues. - Build $D = \text{diag}(\lambda_1, \lambda_2, \ldots)$ and $P$ from normalized eigenvectors. - Compute $A = P D P^T$. 2.i. Newton-Raphson method formula: - Given approximate root $x_n$, the next approximation $x_{n+1}$ is $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$ - Show this formula applies to the given equation and $x_n$. 2.ii. Determine root to 4 decimal places: - Start with initial guess $x_0$. - Use iteration formula to get $x_1$, $x_2$, ... until $$ |x_{n+1} - x_n| < 0.0001 $$ - Final $x_{n+1}$ is root correct to 4 decimal places. 3. Newton-Gregory forward difference interpolation: - Using table of $x$ and $f(x)$ values, compute forward differences \( \Delta f \). - Interpolation formula: $$ f(x) = f(x_0) + \frac{(x - x_0)}{1!} \Delta f(x_0) + \frac{(x - x_0)(x - x_1)}{2!} \Delta^2 f(x_0) + ... $$ - Calculate each forward difference from table data, and plug values for desired $x$. 4.i. Express system in state-space form: - Represent system as $$ \frac{d\mathbf{x}}{dt} = A(t) \mathbf{x} + B(t) u(t) $$ where $\mathbf{x}$ is state vector, $A(t)$ is system matrix, $B(t)$ input matrix. 4.ii. Determine state transition matrix $\Phi(t)$: - Solve $$ \frac{d \Phi}{dt} = A(t) \Phi(t), \quad \Phi(0) = I $$ - Use matrix exponentials or known solution methods for time-varying systems. 5.i. Express $u$ and $v$ in complex form: - Write complex function $f(z) = u(x,y) + i v(x,y)$. 5.ii. Show Cauchy-Riemann equations hold: - Verify $$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $$ - Compute partial derivatives and check equality. Final answers depend on given data but the method steps guide the solution.