I am trying to understand the Schmidt Decomposition, currently in my QC class. We had a tutorial where we were told if $|\psi\rangle$ is a pure state of a composite system A then there exists $|i_A\rangle$ and $|i_B\rangle$ which are orthonomral states for systems A and B respectively.
$$|\psi\rangle = \sum_i \lambda_i |i_A\rangle|i_B\rangle$$
I more or less understand that we write $|\psi\rangle$ as:
$$ |\psi\rangle = \sum_{j,k} c_{jk}|a_j\rangle|b_k\rangle$$
Where $|a_j\rangle_j$ and $|b_k\rangle_k$ are orthonormal bases for A and B and $C=(c_{jk})$ a complex matrix. Then proceed by using singular value decomposition on C. My question is how should I move on to apply the Schmidt decomposition on a numerical example such as the $|\beta_{00}\rangle$. Some pointers would be lovely as I struggle to understand, and would prefer it if there were no direct answers as I want to struggle through it myself as much as possible. Thanks!