Question

Problem 1. Let A1 = 2 −1 −1 2 , A2 = 2 −5 −1 2 . (1) 1. Generate 1000 random values of x. 2. For each x, compute J1 = x TA1x and J2 = x TA2x. 3. Use the scatter3 function to plot J1 and J2 in Matlab. Use the rotate feature of the Matlab figure to deduce whether J1 and J2 have a minimizer or not. 4. Is A1 positive definite

Answer 1

The task requires generating quadratic forms from matrices and plotting them to deduce the existence of minimizers. A1 appears to be positive definite since its leading principal minors are positive.

Step-by-step explanation:

The student's question involves working with matrices and vectors in MATLAB to conduct a mathematical analysis and visualization. Once 1000 random values of x are generated, two quadratic forms J1 and J2 are computed using the matrices A1 and A2 by the formula J = xTAx.

By plotting J1 and J2 using the scatter3 function in MATLAB, one can visually inspect these plots to deduce the existence of minimizers for the quadratic forms. To determine if A1 is positive definite, one must look at the matrix's properties, such as its eigenvalues or use the Sylvester's criterion.

For checking positive definiteness of the matrix A1, one must evaluate if all its leading principal minors are positive. Since A1 is a 2x2 matrix, it's quite straightforward:

The determinant of the 1x1 leading principal minor (the top-left element of the matrix) must be positive, which is (2), and the determinant of the full matrix should also be positive:

det(A1) = (2)(2) - (-1)(-1) = 4 - 1 = 3.

Both conditions are satisfied; hence, A1 is positive definite.