<s>[INST] # Eigenvalue decomposition of degenerate matrix

Assume a vector \( \vec{\mathbf{x}} = [ x_1, x_2, x_3 ]^T \) has been predefined in a n-dimensional space.

For simplicity let's focus on two-dimensional case \( n = 2 \) and assume \( \vec{\mathbf{x}} = [ x_1, x_2 ]^T \).

Given a D-dimensional identity matrix \( \mathbf{I} \), consider the \( \mathbf{I} \):

$$\mathbf{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

Now, for a real symmetric matrix \(\mathbf{A}\) with eigenvalue \(\lambda_1 = 1\), \(\lambda_2\) and eigenvectors \(\vec{\mathbf{v_1}}\) and \(\vec{\mathbf{v_2}}\), the eigenvalue decomposition can be given as \(\mathbf{A} = \mathbf{V}\mathbf{\Lambda}\mathbf{V}^{-1} \).

$$
\mathbf{A} = \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} \vec{\mathbf{v_1}} & \vec{\mathbf{v_2}} \end{bmatrix}\begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}\begin{bmatrix} \vec{\mathbf{v_1}} & \vec{\mathbf{v_2}} \end{bmatrix}^{-1}\\
$$

After calculate the \(\lambda\)s, we found that\(\lambda_2 = 0\). Now how do we handle the inverted (or pseudoinverted) matrices here?[/INST]