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cdf2rdf

Convert complex diagonal form to real block diagonal form

Syntax

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

Description

If the eigensystem [V,D] = eig(X) has complex eigenvalues appearing in complex-conjugate pairs, cdf2rdf transforms the system so D is in real diagonal form, with 2-by-2 real blocks along the diagonal replacing the complex pairs originally there. The eigenvectors are transformed so that

X = V*D/V

continues to hold. The individual columns of V are no longer eigenvectors, but each pair of vectors associated with a 2-by-2 block in D spans the corresponding invariant vectors.

Examples

The matrix

X =
    1     2     3
    0     4     5
    0    -5     4

has a pair of complex eigenvalues.

[V,D] = eig(X)
          
V = 

    1.0000      -0.0191 - 0.4002i     -0.0191 + 0.4002i
         0            0 - 0.6479i           0 + 0.6479i
         0       0.6479                0.6479          

D =

    1.0000            0                     0
         0       4.0000 + 5.0000i           0
         0            0                4.0000 - 5.0000i

Converting this to real block diagonal form produces

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

V =

    1.0000    -0.0191     -0.4002
         0          0     -0.6479
         0     0.6479           0

D =

    1.0000          0           0
         0     4.0000      5.0000
         0    -5.0000      4.0000

More About

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Algorithms

The real diagonal form for the eigenvalues is obtained from the complex form using a specially constructed similarity transformation.

See Also

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