= Solution
The <Schmidt decomposition> of a normalized bipartite pure state is
$$
\boxed{|\psi\rangle_{AB}=\sum_{j=1}^{r}\sqrt{\lambda_j}\,|u_j\rangle_A|v_j\rangle_B},
\qquad \lambda_j>0,\quad\sum_j\lambda_j=1.
$$
The two sets of local vectors are orthonormal. The <Schmidt coefficients> are the nonnegative numbers $\sqrt{\lambda_j}$, and the Schmidt number for this pure state, also called its <Schmidt rank>, is $r$, the number of nonzero coefficients. It equals the rank of either <reduced density operator> and of the coefficient matrix in any product bases. The general mixed-state <Schmidt number> specializes to this same value on a pure state.
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