# 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```