**Definition:** The **singular value decomposition (SVD)** of an $m \times n$ matrix $A$ is a factorization:
$$A = U \Sigma V^*$$
where:
- $U$ is an $m \times m$ unitary (orthogonal if real) matrix
- $V$ is an $n \times n$ unitary (orthogonal if real) matrix
- $\Sigma$ is an $m \times n$ diagonal matrix with non-negative entries $\sigma_1 \geq \sigma_2 \geq \cdots \geq 0$ called **singular values**
**Key facts:**
- Singular values are the square roots of eigenvalues of $A^*A$ (or $AA^*$)
- Columns of $U$ are left singular vectors, columns of $V$ are right singular vectors
- The number of nonzero singular values equals the rank of $A$
- The SVD exists for **every** matrix (real or complex, square or rectangular)
**Applications:** data compression, PCA, pseudoinverse computation, least squares, image processing.