problem1) a) Let M and T be a metro city and a nearby district town, respectively. Our government is trying to develop infrastructure in T so that people shift to T. Each year 10% of T’s population moves to M and 1% of M’s population moves to T. What is the long term effect of this on the populations of M and T? Are they likely to stabilise?
b) The circuit in Figure 1 is given by the matrix equation
where v_{1}(t) and v_{2} (t) are the voltages across the capacitors C_{1} and C_{2} at time t. Suppose the resistor R_{1} is 1 ohm, R_{2} is 2 ohms, C_{1} is 1 farad, C_{2} is 0.5 farad, there is an initial charge of 5 volts on C_{1} and 4 volts on C_{2} . Then solve the equation.
c) Give an ex each to show that the following statements are false.
i) If A_{1} and A_{2} are n × n matrices which are similar to B_{1} and B_{2} , respectively, then A_{1} A_{2} is similar to B_{1} B_{2} .
ii) Two n n ×matrices with the same characteristic polynomial mu st be similar.
problem2) Find a unitary matrix U such that * U AU is upper triangular, where A=
Hence obtain an orthonormal basis for the linear operator T on R^{3} with respect to which the matrix of T is upper triangular. Here T is defined by A w.r.t. the standard orthonormal basis on R^{3}.
problem3) Find the least squares solution to Ax = b, where A=, b=[-4 3]^{t} by
i) using the SVD of A.
ii) solving the system A^{t} Ax = A^{t}b