Singular Value Decomposition

X x_{k} U, V Σ = 1 x_{1} ⋮ 1 1 x_{2} ⋮ x_{1} \dots \dots ⋮ \dots 1 x_{m} ⋮ x_{m} = U Σ V^{T} = 1 u_{1} ⋮ 1 1 u_{2} ⋮ 1 \dots \dots ⋮ \dots 1 u_{n} ⋮ 1 σ_{1} ⋮ ⋮ ⋮ 0 \dots σ_{2} \dots \dots 0 \dots \dots ⋱ \dots 0 \dots \dots \dots σ_{m} 0 ⋮ v_{1} ⋮ ⋮ v_{2} ⋮ ⋮ \dots ⋮ ⋮ v_{n} ⋮^{T} \in R^{n} : unitary matrices : diagonal matrix

where $1 u_{1} ⋮ 1$ are “eigen-faces”

$U$ is orthonormal, meaning:

U U^{T} V V^{T} Σ = U^{T} U = I_{n \times n} = V^{T} V = I_{m \times m} : diagonal σ_{1} \geq σ_{2} \geq \dots \geq σ_{m} \geq 0