수식입력연습

\begin{equation} \displaylines { Q &=& \frac{\text{Displace amplitude at resonance}} {\text{Static deflection due to force amplitude}} \\ &=& \frac{m\omega_0}{R_m} = \frac{1}{2\zeta} = \frac{\omega_0}{\text{Half-power bandwidth}} } \end{equation}

\begin{equation} \displaylines { f\left( x-ct \right) \ \ \ \text{and} \ \ \ g\left( x+ct \right) \\ \text{where f, and g are arbitrary functions} } \end{equation}

\begin{equation} \nabla ^2f = \sum^{n}_{i=1}\frac{\partial ^2 f}{\partial x^2_i} \end{equation}

\begin{equation*} \displaylines { J &=& \Vert A\theta - y \Vert^2_2 + \lambda \Vert \theta \Vert^2_2 \\ &=& \left( A\theta - y \right)^T \left( A\theta - y \right) + \lambda \left( \theta^T \theta \right) \\ &=& \left( \theta^TA^T - y^T \right)^T \left( A\theta - y \right) + \lambda \left( \theta^T \theta \right) \\ &=& \theta^T A^T A \theta - \theta^T A^T y - y^T A \theta + y^T y + \lambda \left( \theta^T \theta \right) \\ \Rightarrow \frac{\partial J}{\partial \theta} &=& A^T A \theta + \left( A^T A \right)^T \theta - A^T y - \left( y^T A \right)^T + 2 \lambda \theta \\ &=& 2 A^T A \theta - 2 A^T y + 2 \lambda \theta = 0 \\ \Rightarrow \left( A^T A + \lambda I_n \right) \theta &=& A^T y \\ \Rightarrow \hat{\theta} &=& \left( A^TA + \lambda I_n \right)^{-1} A^T y } \end{equation*}

\begin{equation*} \displaylines { \theta \leftarrow \theta - \eta \frac{\partial J}{\partial \theta} \\ \theta \leftarrow \theta - 2 \eta \left( A^T A \theta - A^T y + \lambda \theta \right) } \end{equation*}