Question 6: Solve using the state-transition matrix, the state vector x(t) and the output y(t) -2 |x(t);y = |1 3| X;X(0) = ||| -9-

3. A quadratic function in n variables is any function defined on R" which can be expressed in the form f(x) = a + b·x + x• Ax, where a € R, be R", and A is an n × n-symmetric matrix. (a) Show that the function f(x) defined on R² by f(x₁, x₂) = (x₁ - x₂)² + (x₁ + 2x₂ + 1)² − 8x₁x₂ is a quadratic function of two variables by finding the appropriate a € R, b = R², and the 2 x 2-symmetric matrix A. (b) Compute the gradient Vf(x) and the Hessian Hf(x) of the quadratic function in (a) and express these quantities in terms of the a € R, b = R², and the 2 x 2-symmetric matrix A computed in (a). (c) Show that a quadratic function f(x) of n variables is convex if and only if the corresponding n × n-symmetric matrix A is positive semidefinite, and is strictly convex if A is positive definite. (d) If f(x) is a quadratic function of n variables such that the corresponding matrix A is positive definite, show that 0 2Ax + b has a unique solution and that this solution is the strict global minimizer…

The function of 2 variables:  f (x,y) = (5/2)x2 + 2xy - 12x + (3/2)y2 - 7y + (31/2)  Has a stationary point at (2,1). At this stationary point, the Hessian matrix for f is:  H = [5 2]        [2 3]  Which one of these options best describes the stationary point? A. The stationary point is a saddle point B. The stationary point is a local minimum C. The stationary point is a local maximum D. The nature of the stationary point cannot be determined by the Hessian matrix

U and = span{[-3 4 5],[-11 -2 3]} W = span{[ span{[−4 x = [ UnW = −1 5],[-12 -1 1x×3 of V = R¹×³. Find a matrix X € V such that E 5], [-12 -5 9]} = span{X}.