91Ó°ÊÓ

To prove that H is Hermitian, we need to show that H = H^*. Given that H = A^* A, we need to find the conjugate transpose of H: H^* = (A^* A)^* Remembering that (AB)^* = B^* A^*, we have: H^* = A^* (A^*)^* Now, using the property that (A^*)^* = A, we get: H^* = A^* A Which is equal to H. Therefore, H is Hermitian.

To prove that H is positive definite, we need to show that for any non-zero complex vector x, the expression x^* H x > 0. So, let's evaluate x^* H x for a non-zero complex vector x: x^* H x = x^* (A^* A) x Using the associative property, we can rearrange this as: x^* H x = (x^* A^*) (A x) Let y = A x (because A is nonsingular, A x is nonzero when x is nonzero). Then, the expression becomes: x^* H x = y^* y Now, the expression y^* y represents the inner product of y with itself, written as ⟨y, y⟩. By definition, the inner product is always nonnegative for any vector y and equal to zero if and only if y = 0. In our case, since we started with a non-zero x, we know that A x = y ≠ 0 (because A is nonsingular). Therefore, y^* y > 0, which implies that x^* H x > 0. Thus, H is positive definite.