Let z = a + ib and w = c + id, a, b, c, d R. Show that and (the line over a comp

Question

Let z = a + ib and w = c + id, a, b, c, d R. Show that and (the line over a complex number or a vector means its complex conjugate). Use (i) to show that if A Rn times n and mu = alpha + beta, alpha, beta R, is an eigenvalue of A and p = pr + ipi, pr, pi Rn, a corresponding eigenvector, then = alpha - i beta is also an eigenvalue of A and = pr - ipi is a corresponding eigenvector. Is (ii) necessarily true if A Cn times n? (Hint: .) If a A R3 times 3 has the eigenvalues lambda1 = a + ib. Lambda2 = a - ib, and lambda3 = c + id. d b, then show that d = 0. Hint: how many eigenvalues does A have?

Explanation / Answer

i dont think proving part 1 is necessary because it is pretty obvious
now in part 2
we have Ap = p

take conjugate on both the sides we have

(Ap)' = (p)' ; where x' dentes conjugate of x

= Ap' = p'' (since A is real)

so ' is also eigen value of A with eigen vector p'

but if A is complex then A' is not equal to A hence above proof is not valid

but however ' is eigen value of A'

now given that 1,2,3 are eigen values of A

dimension of A =3 and hence it has only 3 eigen values

now from part 2 we have 1',2',3' are also eigen values of A

although 1'=2 we have an extra 3' for an eigen value

so 3 must be real, only then 3=3'

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