What the Rank-One Copy Remembers

A tiny search index stores how often each of three terms appears in each of two documents:

As a term-by-document matrix,

$A = \begin{bmatrix} 3 & 2 \\ 2 & 0 \\ 0 & 2 \end{bmatrix}$

Compressing this index means writing $A = U\Sigma V^{\top}$ and keeping only the largest singular value — the single latent topic that explains the most of the data. That gives the best rank-one approximation

$A_1 = \sigma_1\,\mathbf{u}_1\mathbf{v}_1^{\top}$

where $\mathbf{u}_1$ and $\mathbf{v}_1$ are the unit-length left and right singular vectors belonging to $\sigma_1$, and $\mathbf{u}_1$ is taken with the sign that makes $A\mathbf{v}_1 = \sigma_1\mathbf{u}_1$.

$A_1$ is the count table the compressed index would reconstruct. You want to know what it reconstructs for term 2 in document 1 — the entry of $A_1$ in row 2, column 1, where the original count was $2$.

What is the entry of $A_1$ in row 2, column 1? Round your answer to 3 decimal places.