What's the detailed derivation from equation 126 to 127 (See Figure)? (problem of Lagrange method with vector transpose)?

Question

I am confused of vector transpose, I do not know how to get equation 127. Thanks.

Answer 1

$$-\sum_{i=1}^m\left(y_i-\mathbf{w}^T\mathbf{x_i}\right)\mathbf{x_i}+\lambda\mathbf{w}=0$$

$$\Longleftrightarrow -\left(\sum_{i=1}^my_i\mathbf{x_i}\right)+\left(\sum_{i=1}^m\mathbf{x_i}\mathbf{x_i}^T\right)\mathbf{w}+\lambda\mathbf{w}=0$$

$$\Longleftrightarrow\left(\sum_{i=1}^m\mathbf{x_i}\mathbf{x_i}^T+\lambda\mathbf{I}\right)\mathbf{w}=\sum_{i=1}^my_i\mathbf{x_i}$$

$$\Longleftrightarrow\mathbf{w}=\left(\sum_{i=1}^m\mathbf{x_i}\mathbf{x_i}^T+\lambda\mathbf{I}\right)^{-1}\left(\sum_{j=1}^m\mathbf{x_j}y_j\right)$$