Question

5. Recall that a symmetric matrix A is positive definite (SPD for short) if and only if T Ar > O for every nonzero vector 2. 5a. Find a 2-by-2 matrix A that (1) is symmetric, (2) is not singular, and (3) has all its elements greater than zero, but (1) is not SPD. Show a nonzero vector such that zAx < 0. 5b. Let B be a nonsingular matrix, of any size, not necessarily symmetric. Prove that the matrix A= BTB is SPD.

Answer 1

@ take Aa 27 them on yo fe 3.1987 [xtzy_22+41 f47 = n²+2uytzuyt ya = u²t2ay + y²tany x An = (uty?tany where u. [u y] now if 2:11-17 teen xaua-250 6 for panitive dyinib matrix A, x'Auzo should setisfy for all non-7070 u. taking A- BTB, we have kanzur B x= (Bu) (Bu) = 11 Bull?. This squared length is pasitive, unless u-o, becaus B is mvertible hance Columns of B are linearly Independent. This B B is pasitive definite