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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 ```

Algorithms

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