For an $m \times n$ matrix $A$, the SVD is a factorization:

$$ A = U\Sigma V^T $$

Components:

• $U$ is an $m \times m$ orthogonal matrix (left singular vectors)
• $\Sigma$ is an $m \times n$ diagonal matrix (singular values $\sigma_1 \geq \sigma_2 \geq \cdots \geq 0$)
• $V^T$ is an $n \times n$ orthogonal matrix (right singular vectors)

Key Properties:

• Singular values are the square roots of eigenvalues of $A^TA$ (or $AA^T$)
• The rank of $A$ equals the number of non-zero singular values
• SVD works for any matrix (square or rectangular)