If the entries of the vector z z1 zmt are real-valued

Question

Question: Let z = ||Ax - b||2. Show that

(∂z/∂x) = 2A^TAx - 2A^Tb.

Use z = (Ax - b)T (Ax - b).

We can also consider the second derivative of z = f(x), which is the Hessian matrix of z with entries

H = ∂²z/∂x² = ∇²f(x)

H_mn = ∂²z/∂x_m∂x_n

If the entries of the vector z = (z1, ..., zM)^T are real-valued functions of the vector x, the derivative of z is the matrix whose mth column is the derivative of the real-valued function zm. This matrix is usually called the Jacobian matrix of z. If M = N the determinant of the Jacobian matrix is the Jacobian.