Let Pk denote a rotation matrix of the form given in (9.17). a. Show that Pt2 Pt3

Question:

Let Pk denote a rotation matrix of the form given in (9.17).
a. Show that Pt2 Pt3 differs from an upper triangular matrix only in at most the (2, 1) and (3, 2) positions.
b. Assume that Pt2 Pt3· · · Ptk differs from an upper triangular matrix only in at most the (2, 1), (3, 2), . . . , (k, k−1) positions. Show that Pt2 Pt3 · · · Ptk Ptk +1 differs from an upper triangular matrix only in at most the (2, 1), (3, 2), . . . , (k, k − 1), (k + 1, k) positions.
c. Show that the matrix Pt2 Pt3 · · · Ptn is upper Hessenberg.