SingularValues - Maple Help

Student[LinearAlgebra]

 SingularValues
 compute the singular values of a Matrix

 Calling Sequence SingularValues( A, options )

Parameters

 A - Matrix, list; Matrix whose singular values are required

Options

 • options: parameters; for a complete list, see LinearAlgebra[SingularValues]

Description

 • The SingularValues(A) function returns the singular values of Matrix A in a column Vector. The singular values are equal to the square roots of the (real) eigenvalues of $A·\mathrm{Transpose}\left(A\right)$ or $A·\mathrm{HermitianTranspose}\left(A\right)$ (see below). Since this product is either real-symmetric or Hermitian, and positive semi-definite, the eigenvalues are all real and non-negative, and so their square roots are also purely real.
 • When calling SingularValues(A) with output=['U','S','Vt'], the Matrix U, Vector S, and Matrix Vt satisfy $U·T·\mathrm{Vt}=A$, where $T=\mathrm{DiagonalMatrix}\left(S\right)$.
 • The values of hardwarefloats (default false) and conjugate (default false) for the Student package can be set using the Student[SetDefault] command.
 • If the Matrix has an element that is not of type complex(numeric), the value of conjugate will be used to deduce if HermitianTranspose should be used. On the other hand, if all the elements are of type complex(numeric), then the regular transpose will be used if the elements are real-valued, and the Hermitian transpose if the elements are complex-valued.
 • For further information, see LinearAlgebra[SingularValues].

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right):$

Example 1

 > $A≔\mathrm{Matrix}\left(\left[\left[1.0,2.0\right],\left[3.0,4.0\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{cc}{1.0}& {2.0}\\ {3.0}& {4.0}\end{array}\right]$ (1)
 > $S≔\mathrm{SingularValues}\left(A\right)$
 ${S}{≔}\left[\begin{array}{c}{5.464985704}\\ {0.3659661916}\end{array}\right]$ (2)

Example 2

 > $B≔\mathrm{Matrix}\left(\left[\left[4.575068354,-4.024595950,-3.730131837\right],\left[0.4688151920,1.323592462,4.057919371\right],\left[-2.215017811,4.133758561,3.147236864\right]\right]\right)$
 ${B}{≔}\left[\begin{array}{ccc}{4.575068354}& {-4.024595950}& {-3.730131837}\\ {0.4688151920}& {1.323592462}& {4.057919371}\\ {-2.215017811}& {4.133758561}& {3.147236864}\end{array}\right]$ (3)
 > $U,S,\mathrm{Vt}≔\mathrm{SingularValues}\left(B,'\mathrm{output}'=\left['U','S','\mathrm{Vt}'\right]\right)$
 ${U}{,}{S}{,}{\mathrm{Vt}}{≔}\left[\begin{array}{ccc}{-0.7381898079}& {-0.4056519114}& {0.5390012395}\\ {0.3306799655}& {-0.9140152493}& {-0.2350040101}\\ {0.5879851780}& {0.004759346363}& {0.8088577004}\end{array}\right]{,}\left[\begin{array}{c}{9.482762013}\\ {3.196679524}\\ {1.112982108}\end{array}\right]{,}\left[\begin{array}{ccc}{-0.4771435440}& {0.6157689050}& {0.6270268681}\\ {-0.7179110309}& {0.1384171711}& {-0.6822348853}\\ {0.5068903140}& {0.7756734764}& {-0.3760224302}\end{array}\right]$ (4)
 > $T≔\mathrm{DiagonalMatrix}\left(S\right)$
 ${T}{≔}\left[\begin{array}{ccc}{9.482762013}& {0.}& {0.}\\ {0.}& {3.196679524}& {0.}\\ {0.}& {0.}& {1.112982108}\end{array}\right]$ (5)
 > $\mathrm{Norm}\left(U·T·\mathrm{Vt}-B\right)$
 ${1.115765771}{×}{{10}}^{{-8}}$ (6)

Compatibility

 • The Student[LinearAlgebra][SingularValues] command was introduced in Maple 2021.