Problem: Eigenvalue Analysis and Differential Equations in a Topological Space Consider the following setup: 1. Let A = Rnxn be a real-valued, symmetric matrix, and let A1, A2, ..., An be its eigenvalues, with corresponding orthonormal eigenvectors V1, V2, ..., Un 2. Define a linear system of ordinary differential equations (ODES): dx(t) dt = AX(t) where x(t) € R" is a time-dependent vector. The initial condition is X(0) = Xo, where X₁ Є R". 3. Now, consider a topological space T, a compact subset of R", and let f: TT be a continuous mapping defined by the solution to the ODE: f(x) = x(to) for some fixed time to > 0. Part 1: Spectral Decomposition and Solution to the ODE dx(t) = dt AX (t) in terms of the ⚫ (a) Use the spectral theorem to decompose the matrix A in terms of its eigenvalues and eigenvectors. Express the general solution X (t) of the system eigenvalues A1, A2,..., An and the initial condition Xo. • (b) Show how the solution behaves as t→ ∞ depending on the signs of the eigenvalues A1, A2,..., An Part 2: Stability Analysis and Critical Points • (a) Identify the critical points of the ODE system. Use the eigenvalue information to classify the stability of these critical points. (b) Assuming ₁ > 0 and A2,..., An <0, analyze the long-term behavior of the solutions and the direction in which the trajectories of the system evolve. Part 3: Topological Dynamics and Brouwer's Fixed Point Theorem ⚫ (a) Consider TCR as a compact and convex set. Show that, by Brouwer's Fixed Point • Theorem, the continuous map f has at least one fixed point in T. (b) If the matrix A has both positive and negative eigenvalues, describe the implications for the dynamical system in terms of the topology of the trajectories within T. ⚫ (c) What can you say about the uniqueness of the fixed point if A is not positive definite? Discuss the impact of the sign of the eigenvalues on the topological structure of the fixed points. Part 4: Numerical Methods and Simulation ⚫ (a) Propose a numerical method (e.g., Euler's method, Runge-Kutta, etc.) for approximating the solution to the ODE system. Write down the algorithm for implementing the numerical method. (b) Analyze the stability of the numerical method in terms of the eigenvalues A1, A2, ..., An of the matrix A. For which eigenvalues does the method become unstable? (c) Implement a simulation for a 2x2 matrix A, where A₁ = 2 and λ₂ = -1. Plot the trajectories of the system over time and analyze the long-term behavior. Part 5: Functional Analysis Perspective • • (a) Consider the operator L : C([0, to], R²) → C([0, to], R") defined by the ODE dx(t) AX (t). Show that L is a bounded linear operator and compute its operator norm. (b) Determine whether the operator L is compact. If so, prove this fact. If not, provide a counterexample. = (c) If A has distinct eigenvalues, describe the spectral properties of the operator L, and discuss its implications for the behavior of solutions to the ODE system.