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# bdiag

block diagonalization, generalized eigenvectors

### Syntax

`[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

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

• schur — [ordered] Schur decomposition of matrix and pencils
• sylv — Sylvester equation.
• spec — eigenvalues of matrices and pencils
• blockdiag — Создаёт блочную диагональную матрицу из указанных массивов. Соединение диагональной системы блоков.

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