For matrix $X\in\mathbb{R}^{m\times n}$, rank is the dimension of its column space, equivalently its row space. If $\operatorname{rank}(X)=r$, then all columns are combinations of only $r$ independent directions. Rank cannot exceed $\min(m,n)$ and can reveal redundancy hidden inside a large array of numbers.
Any rank-$r$ matrix can be factored as $X=AB$ with $A\in\mathbb{R}^{m\times r}$ and $B\in\mathbb{R}^{r\times n}$. Storing the factors uses $r(m+n)$ values instead of $mn$. Compression occurs only when $r(m+n)<mn$, and multiplication can exploit the factors without reconstructing the full matrix.
A singular decomposition writes $$X=U\Sigma V^T,$$ where columns of $U$ and $V$ are orthonormal and diagonal entries $\sigma_1\geq\sigma_2\geq\cdots\geq0$ are singular values. The number of nonzero singular values equals rank. Each term $\sigma_i u_iv_i^T$ is a rank-one contribution.
Keeping only the largest $k$ singular values gives $$X_k=U_k\Sigma_kV_k^T.$$ Among all rank-$k$ matrices, this choice minimizes the sum of squared entrywise reconstruction errors. The squared error is $\lVert X-X_k\rVert^2=\sum_{i>k}\sigma_i^2$, so the singular-value tail quantifies exactly what the truncation discards.
Low rank is an approximation assumption, not a universal property. A rapidly decaying spectrum means a small $k$ retains most squared energy; a flat spectrum means aggressive truncation loses substantial information. Task performance may depend on a low-energy direction, so energy retention alone does not guarantee preserved predictions.
Applications include compressed weight updates, embedding analysis, denoising, and reduced computation. Report chosen rank, parameter count, reconstruction error, and task metrics. Numerical rank also depends on tolerance: tiny singular values may be nonzero mathematically but unusable under finite precision or noisy data.