hi lets assume we have matrix A with 4 rows x 3 columns, when we input to svd it become:
U = 4x4 , S = 4x3 and VT = 3x3.
after that i specify that rank k=2 so what exactly happen to the other than the first k ranks? is it set to zero or completely gone like
U = 4x2 S=2x2 and VT = 2x3?
both cases resulting the same rows and columns, right? which is 4x3, but does it affect the value of the reduced matrix?
In your case, $A = USV^T$, where $$U = [u_1 \ u_2 \ u_3 \ u_4]$$ $$V = [v_1 \ v_2 \ v_3]$$ where $u_i$ is the $i^{th}$ column of $U$. (Same for $V$). Now, $S$ is a diagonal rectangular containing the Singular values of $A$, i.e. $$S = \begin{bmatrix} \sigma_1 & 0 & 0 \\ 0 & \sigma_2 & 0 \\ 0 & 0 & \sigma_3 \\ 0 & 0 & 0 \\ \end{bmatrix}$$ Notice that the number of non-zero singular values indicates the rank of $A$, hence if you have rank$A$ = 2, then (assuming the singular values are sorted $\sigma_1 \geq \sigma_2 \geq \sigma_3$), we will have $\sigma_3 = 0$ Hence $$S = \begin{bmatrix} \sigma_1 & 0 & 0 \\ 0 & \sigma_2 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \\ \end{bmatrix}$$ Hence $$A = USV^T = \sigma_1 u_1v_1^T + \sigma_2 u_2v_2^T = \bar{U}\bar{S}\bar{V}^T$$, where $$\bar{U} = [u_1 \ u_2] \in \mathbb{R}^{4\times2}$$ $$\bar{V} = [v_1 \ v_2 ] \in \mathbb{R}^{3\times2} $$ and $$\bar{S} = \begin{bmatrix} \sigma_1 & 0 \\ 0 & \sigma_2 \\ \end{bmatrix} \in \mathbb{R}^{2\times2}$$