I only understand the singular value decomposition process. Do I apply it to this matrix?
\begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix}
What is the idea behind matrix approximation using this process?
Applying SVD, we have $\begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix}^T\begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & e^2 & 0 \\ 0&0&\pi^2 \end{bmatrix}$
How to proceed?
Best Answer
The idea is as follows: we find the SVD of this matrix, which has the form $$ A = U \Sigma V^T = \pmatrix{u_1 & u_2 & u_3} \pmatrix{\sigma_1 \\&\sigma_2\\&& \sigma _3} \pmatrix{v_1^T\\v_2^T \\ v_3^T} = \sum_{i=1}^3\sigma_i u_i v_i^T $$ The rank-2 approximation is $$ A' = \pmatrix{u_1 & u_2 & u_3} \pmatrix{\sigma_1 \\&\sigma_2\\&& 0} \pmatrix{v_1^T\\v_2^T \\ v_3^T} = \sum_{i=1}^2\sigma_i u_i v_i^T $$