DO 3. 24 pts) Let , v R" be given linearly independent unit vectors and let w = 8 + . Define the matrix P= In - 2(w w) - T + 2y Show that (a) P=UJU, where U = In - 2 (Two) - ww", Uz = In - 25z are two Householder matrices; (b) det P = 1; (c) P= y and Py= - +257); (d) P= { if ze R", = 0, and 7 y=0. (e) The eigenvalues of P are i" u + iv1 - (577)2, T- ivi - (TT)2, 1,...,1. DO 3. 24 pts) Let , v R" be given linearly independent unit vectors and let w = 8 + . Define the matrix P= In - 2(w w) - T + 2y Show that (a) P=UJU, where U = In - 2 (Two) - ww", Uz = In - 25z are two Householder matrices; (b) det P = 1; (c) P= y and Py= - +257); (d) P= { if ze R", = 0, and 7 y=0. (e) The eigenvalues of P are i" u + iv1 - (577)2, T- ivi - (TT)2, 1,...,1