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 $U\Sigma V$ where $U$ is any square real or complex Unitary matrix of order $mxm$.
• $\Sigma$ 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 $M^{*}M$.
*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 $M^{*}M$ and $MM^{*}$.