# 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 — [ordered] Schur decomposition of matrix and pencils
- sylv — Solve the Sylvester equations A*X + X*B = C or A*X*B + X = C
- spec — eigenvalues, and eigenvectors of a matrix or a pencil
- blockdiag — Creates a block diagonal matrix from provided arrays. Block diagonal system connection.
- lcmdiag — least common multiple diagonal factorization

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