. Using column—row multiplication, show that any n by n symmetric matrix S = Q A QT

splits into n rank one pieces : S=A1qqu +—+/\nqnq$ For which matrices S is this the same as the SVD of S? . Suppose Q is a 6 by 4 matrix with orthonormal columns ql to (14. Show that QTQ = I.

Show that QQT does not equal I. . Suppose S = Q A QT is symmetric.

Show that S2 = Q A2 QT. So, 82 has the same as 8. What is the test to decide if

S approaches zero for n —) 00? . Find the minimum value and the (3:1, 332) that produces this minimum for the function 1

f(x) = §$T3$ — 4331 — 2×2 :

_ 5 4 _ 1’1

3- l4 5i 1’- [ml

. Suppose n by n matrices A and B have the same n independent eigenvectors (with

different eigenvalues). Show that AB = BA. . Describe all matrices M that are both orthogonal and symmetric. . Explain why a positive deﬁnite symmetric matrix 3 cannot have S: = 0 in the top left

corner. The energy test is often the best. Math