Main Content

cdf2rdf

Convert complex diagonal form to real block diagonal form

所有的页面崩溃

Syntax

[Vnew,Dnew] = cdf2rdf(V,D)

Description

example

[Vnew,Dnew] = cdf2rdf(V,D)transforms the outputs of[V,D] = eig(X)or[V,D] = eigs(X,___)for real matricesXfrom complex diagonal form to real diagonal form. This operation transforms how the eigenvalues ofXare expressed inD, and transformsVsuch thatX*Vnew = Vnew*Dnew.

  • Incomplex diagonal form,Dis a diagonal matrix with complex conjugate pairs of eigenvalues on the main diagonal:

    [ λ 1 a + b i a − b i c + d i c − d i ⋱ ]

    Some of the eigenvalues along the diagonal might be real, but complex conjugate eigenvalue pairs are assumed to be next to one another.

  • Inreal diagonal form,Dnew有真正eigenvalues on the diagonal, and complex eigenvalues are expressed as 2-by-2 real blocks along the main diagonal:

    [ λ 1 a b − b a c d − d c ⋱ ]

Examples

collapse all

Open Live Script

Calculate the eigenvalues and eigenvectors of a real matrix, and then transform the complex conjugate eigenvalue pairs into real blocks.

Find the eigenvalues and eigenvectors of a real matrix.

X = [1 1 1 1 1 0 4 5 1 1 0 -5 4 1 1 0 0 2 3 1 0 0 -3 -2 1]; [V,D] = eig(X)
V =5×5 complex1.0000 + 0.0000i -0.0179 - 0.1351i -0.0179 + 0.1351i 0.1593 - 0.4031i 0.1593 + 0.4031i 0.0000 + 0.0000i 0.0130 - 0.6214i 0.0130 + 0.6214i 0.0704 - 0.0267i 0.0704 + 0.0267i 0.0000 + 0.0000i 0.6363 + 0.0000i 0.6363 + 0.0000i -0.1261 + 0.1032i -0.1261 - 0.1032i 0.0000 + 0.0000i 0.1045 - 0.2087i 0.1045 + 0.2087i -0.2279 - 0.4161i -0.2279 + 0.4161i 0.0000 + 0.0000i -0.1156 + 0.3497i -0.1156 - 0.3497i 0.7449 + 0.0000i 0.7449 + 0.0000i
D =5×5 complex1.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 3.8801 + 5.1046i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 3.8801 - 5.1046i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 2.1199 + 0.7018i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 2.1199 - 0.7018i

Dcontains one real eigenvalue and two pairs of complex conjugate eigenvalues.

TransformVandDso thatDnewis in real block diagonal form andVnewsatisfiesX*Vnew = Vnew*Dnew

[Vnew,Dnew] = cdf2rdf(V,D)
Vnew =5×51.0000 -0.0253 -0.1911 0.2253 -0.5701 0 0.0184 -0.8789 0.0996 -0.0378 0 0.8999 0 -0.1784 0.1459 0 0.1478 -0.2951 -0.3222 -0.5885 0 -0.1634 0.4946 1.0534 0
Dnew =5×51.0000 0 0 0 0 0 3.8801 5.1046 0 0 0 -5.1046 3.8801 0 0 0 0 0 2.1199 0.7018 0 0 0 -0.7018 2.1199

Dnewstill has the real eigenvalue, but the complex conjugate eigenvalues are replaced with 2-by-2 blocks.

Input Arguments

collapse all

Right eigenvector matrix, specified as the matrix returned by[V,D] = eig(X)or[V,D] = eigs(X,___).

Data Types:single|double
Complex Number Support:Yes

Diagonal eigenvalue matrix, specified as the matrix returned by[V,D] = eig(X)or[V,D] = eigs(X,___). Some of the eigenvalues along the diagonal ofDmight be real, but complex conjugate eigenvalue pairs are assumed to be next to one another.

Sinceeigsreturns a subset of the eigenvalues and eigenvectors, the requested number of eigenvalues might include half of a complex conjugate pair.cdf2rdfreturns an error if theDinput contains incomplete complex conjugate pairs.

Data Types:single|double
Complex Number Support:Yes

Output Arguments

collapse all

Transformed right eigenvector matrix, returned as a matrix. The eigenvectors inVare transformed so thatX*Vnew = Vnew*Dnewholds. If the input eigenvector matrixVis unitary, thenVnewis as well. After the transformation, the individual columns ofVneware no longer eigenvectors ofX, but each pair of vectors inVnewassociated with a 2-by-2 block inDnewspans the corresponding invariant vectors.

Transformed diagonal eigenvalue matrix, returned as a block diagonal real matrix. Complex conjugate eigenvalue pairs inDare replaced with 2-by-2 real blocks along the diagonal inDnew.

Extended Capabilities

版本历史

Introduced before R2006a

expand all

See Also

||