Posted: December 30th, 2021

. Using column—row multiplication, show that any n by n symmetric matrix S = Q A QTsplits 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 ifS” approaches zero for n —) 00? . Find the minimum value and the (3:1, 332) that produces this minimum for the function 1f(x) = §$T3$ — 4331 — 2×2 :_ 5 4 _ 1’13- l4 5i 1’- [ml. Suppose n by n matrices A and B have the same n independent eigenvectors (withdifferent eigenvalues). Show that AB = BA. . Describe all matrices M that are both orthogonal and symmetric. . Explain why a positive de?nite symmetric matrix 3 cannot have S“: = 0 in the top leftcorner. The energy test is often the best.

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