Singular value decomposition

SVD is a factorization of a real or complex matrix. It has many useful applications in signal processing and statistics.
Formally, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form UΣV∗ .
U is an m×m real or complex unitary matrix.
Σ is an m×n rectangular diagnal matrix with non-negative real numbers on the diagnal.
V is an n×n real or complex unitary matrix.
the diagnal entries Σi,i of Σ are known as the singular values of M .
the left-singular vectors: columns of matrix U .
the right-singular vectors: columns of matrix V .
Wikipedia https://en.wikipedia.org/wiki/Singular_value_decomposition

unitary matrix : a complex square matrix U is unitary if its conjugate transpose U∗ is also its inverse — that is, if

U∗U=UU∗=I,

where I is the identity matrix.
U∗ is the conjugate transpose of matrix U .

identity matrix: is the n×n square matrix with ones on the main diagnal and zeros else where.

the left-singular vectors of M are a set of orthonormal eigenvectors of MM∗ :
M=UΣV∗
⇒MM∗=(UΣV∗)(UΣV∗)∗
⇒MM∗=UΣV∗V(UΣ)∗
⇒MM∗=UΣΣ∗U∗
⇒MM∗U=UΣΣ∗U∗U
⇒MM∗U=UΣΣ∗