8.[10 marks]8a.[3 marks]Find the lengths and the inner productx·yof the following complexvectorsx=•2-4i4i‚,y=•24‚(i2=-1).8b.[3 marks]LetA=•11-i1 +i2‚. Letx1,x2be two (linearly independent) eigen-vectors ofA. Computex1·x2and show that det(A)∈R.8c.[4 marks]Prove that for any complex vectorxxHAx∈R.(H= Hermitian)Sol.[8a.] length(x) = (xHx)1/2= (£2 + 4i-4i/·•2-4i4i‚)1/2= 6; length(y) = (yTy)1/2=√20andx·y:=xHy= 4(1-2i).[8b.] Notice thatA=AH, furthermore letλibe the 2 eigenvalues ofA: 0 =λ16=λ2= 3,thenx1·x2= 0. Also, one knows that every eigenvalue of a Hermitian matrix is real and sowill be its determinant (det(A) =λ1λ2= 0).[8c.] We have (xHAx)H=xHAx, asA=AH. It follows thatxHAx∈R.