Let's solve the problem step by step: ### **Given** Differentiate the matrix expression \( X^T X \) with respect to \( X \). - \( X \) is a **vector** (let's assume it's an \( n \times 1 \) column vector). - \( X^T X \) is a **scalar** (since \( X^T \) is \( 1 \times n \), and \( X \) is \( n \times 1 \)). --- ## **Step 1: Write the Expression** \[ f(X) = X^T X \] --- ## **Step 2: Expand the Expression** \[ X^T X = \sum_{i=1}^n X_i^2 \] --- ## **Step 3: Differentiate with Respect to \( X \)** Differentiate \( f(X) \) with respect to \( X \): \[ \frac{\partial}{\partial X} (X^T X) \] Since \( X^T X = \sum_{i=1}^n X_i^2 \), its derivative with respect to \( X \) is: \[ \left[ \frac{\partial}{\partial X_1} (X^T X),\ \frac{\partial}{\partial X_2} (X^T X),\ \ldots, \frac{\partial}{\partial X_n} (X^T X) \right]^T = 2X \] --- ## **Step 4: Matrix Calculus Result** In matrix calculus, the result is: \[ \frac{\partial}{\partial X} (X^T X) = 2X \] --- ## **Final Answer** \[ \boxed{2X} \] --- **Explanation:** - \( X^T X \) is a scalar function of the vector \( X \). - Differentiating \( X^T X \) with respect to \( X \) gives \( 2X \).

The final answer is: \[ \boxed{2X} \]

The derivative of \( X^T X \) with respect to \( X \) is \( \boxed{2X} \).