Simplify radical expressions involving fractions

Full solution

\min _{w}\|X w-y\|_{2}^{2}

Identify Meaning: Identify the expression's meaning. $\newline$ The expression $\min_{\mathbf{w}}\|\mathbf{Xw}-\mathbf{y}\|_{2}^{2}$ represents the minimization of the squared Euclidean norm of the vector difference $\mathbf{Xw} - \mathbf{y}$ , where $\mathbf{X}$ is a matrix, $\mathbf{w}$ is a vector of variables, and $\mathbf{y}$ is a vector. This is a common objective in least squares problems.
Rewrite Expression: Rewrite the expression in a more familiar form. $\newline$ The squared Euclidean norm $||\mathbf{Xw} - \mathbf{y}||_2^2$ can be expanded to $(\mathbf{Xw} - \mathbf{y})^T(\mathbf{Xw} - \mathbf{y})$ , which is a standard form in optimization problems involving least squares.
Expand Expression: Expand the expression. $\newline$ Using matrix multiplication, expand $(Xw - y)^T(Xw - y)$ to $w^T X^T X w - w^T X^T y - y^T X w + y^T y$ . Note that $w^T X^T y$ and $y^T X w$ are scalars and equal, so this simplifies to $w^T X^T X w - 2y^T X w + y^T y$ .
Identify Objective: Identify the objective of minimization. $\newline$ To find the minimum of $w^T X^T X w - 2y^T X w + y^T y$ , we need to take the derivative with respect to $w$ and set it to zero. This gives us the normal equations in the context of least squares: $X^T X w = X^T y$ .
Solve for $w$ : Solve for $w$ . Assuming $X^T X$ is invertible, solve the equation $X^T X w = X^T y$ by multiplying both sides by the inverse of $X^T X$ , leading to $w = (X^T X)^{-1} X^T y$ . This is the solution that minimizes the original expression.