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 real or complex matrix is a factorization of the form .

is an real or complex unitary matrix.

is an rectangular diagnal matrix with non-negative real numbers on the diagnal.

is an real or complex unitary matrix.

the diagnal entries of are known as the singular values of .

the left-singular vectors: columns of matrix .

the right-singular vectors: columns of matrix .

Wikipedia https://en.wikipedia.org/wiki/Singular_value_decomposition

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

where is the identity matrix.

is the conjugate transpose of matrix .

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

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the left-singular vectors of are a set of orthonormal eigenvectors of :