Manuals/calci/SVF

SVF (Matrix)

• $Matrix$ is any set of values.

Description

• This function shows the Singular value of a given matrix in descending order.
• In $SVF(Matrix)$, $Matrix$ is any matrix with array of values.
• Singular value decomposition is defined by the factorization of a real or complex matrix.
• It is the generalization of the Eigen decomposition of a symmetric matrix with positive eigen values to any mxn matrix through an extension of the polar decomposition.
• Singular value decomposition is of the form 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 U \Sigma V} where 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 U} is any square real or complex Unitary matrix of order 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 mxm} .
• 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 \Sigma } is a mxn rectangular diagonal matrix with non negative real numbers.
• V is also any square real or complex Unitary matrix of order nxn.
• The columns of U and V are called left Singular and right Singular vectors of the matrix.
• To find Singular Value Decomposition we have to follow the below rules:

*The left-singular vectors of the matrix M are a set of orthonormal eigenvectors of MM∗.
*The right-singular vectors of M are a set of orthonormal eigenvectors of 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 M^*M}
.
*The non-zero singular values of M (found on the diagonal entries of Σ) are the square roots of the non-zero eigenvalues of both 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 M^*M}
 and 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 MM^*}
.

See Also

• LUDECOMPOSITION
• CHOLESKYFACTORIZATION
• QRDECOMPOSITION

References

• Decomposition

• List of Main Z Functions

• Z3 home