bdiag block diagonalization, generalized eigenvectors

## Syntax

                [Ab, X, bs] = bdiag(A)
                [Ab, X, bs] = bdiag(A, rmax)

## Arguments

A

  real or complex square matrix

rmax

  real number

Ab

  real or complex square matrix

X

  real or complex non-singular matrix

bs

  vector of integers

## Description

[Ab, X, bs] = bdiag(A, rmax) performs the block-diagonalization of matrix `A`. `bs` gives the structure of the blocks (respective sizes of the blocks). `X` is the change of basis i.e. `Ab = inv(X)*A*X` is block diagonal.

`rmax` controls the conditioning of `X`. The default value is the l1 norm of `A`.

To get a diagonal form (if it exists), choose a large value for `rmax` (`rmax=1/%eps` for example). Generically (for real random A) the blocks are (1x1) and (2x2) and `X` is the matrix of eigenvectors.

## Examples

    // Real case: 1x1 and 2x2 blocks
    a = rand(5,5);
    [ab, x, bs] = bdiag(a);ab

    --> [ab, x, bs] = bdiag(a);ab
     ab  =
       2.4777836   0.          0.          0.         0.
       0.          0.0757087   0.4869251   0.         0.
       0.         -0.5777956  -0.1248605   0.         0.
       0.          0.          0.          0.069654   0.
       0.          0.          0.          0.         0.5341598

    // Complex case: complex 1x1 blocks
    [ab, x, bs] = bdiag(a+%i*0); ab

## See also

- schur
- sylv
- spec
- blockdiag
- lcmdiag