Manuals/calci/SVD

SVD(Matrix)

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Description

• The singular value decomposition of a matrix A is the factorization of A into the product of three matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = USV^T}
• Where the columns of U and V are orthonormal and the matrix S is diagonal with positive real entries
• Singular value decomposition is defined for all matrices (rectangular or square).

Suppose A is a m × n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers.

• Then there exists a factorization, called a singular value decomposition of A, of the form

where
U is an m × m unitary matrix,
S is a diagonal m × n matrix with non-negative real numbers on the diagonal,
V is an n × n unitary matrix over K, and
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 is the conjugate transpose of V.