Question

Complex Conjugate of Matrix

Answer 1

a^(*)b^(*) = (ab)^(*)

a^(*)+b^(*) = (a+b)^(*)

Matrix multiplication is a sum of products, so A^(-)v^(-) = (Av)^(-) and (A^(-)v)^(-) = (A^(-)^(-)v^(-)) = Av^(-).

(Av)[i] = ∑\_k A[i,k]v[k]

(A^(-)v^(-))[i] = ∑\_k A^(-)[i,k]v^(-)[k]

 = ∑\_k A[i,k]^(*)v[k]^(*)

 = ∑\_k (A[i,k]v[k])^(*)

 = (∑\_k A[i,k]v[k])^(*)

 = (Av)^(-)