- 2 For Any A B Rs1 Define A B C2 X C2 C By Setting X 1 1 0 11 Yi 11 22 Ec 42 Ax141 X1y2 X2y 1 (47.69 KiB) Viewed 19 times
Find a 2 x 2 matrix PA,B such that for all x, y eCa, (x,y)A, B =
y* PA,BX,
Suppose that A, B e R$1. Prove that for all x, y € C, Axī + xy + Ty
+ Byy e R30. Hint: there are a, b, c, d e R such that x = a + bi
and y = c+di; substitute into the above expression, then simplify
and remember that (a + c)2, (b+ d)2 are positive
2. For any A, B € Rs1, define (;-/A,B:C2 x C2 → C by setting X - (1:1) 0:). 11 Yi [] [ 11 22 EC? 42 = Ax141 + x1y2 + x2Y1 + Bx242 for all A,B Show that for any A, B &R>1, the function (; -/A,B is an inner product on C2. Hint: Find a 2 x 2 matrix PA,B such that for all x, y eC2, (x,y)A,B = y* PA,BX, >1 E = then use the previous exercise to show that PA,B is positive definite.