block diagonalization, generalized eigenvectors
[Ab, X, bs] = bdiag(A) [Ab, X, bs] = bdiag(A, rmax)
real or complex square matrix
real number
real or complex square matrix
real or complex non-singular matrix
vector of integers
[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.
// 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 a = rand(5, 5); [ab, x, bs] = bdiag(a+%i*0); ab | ![]() | ![]() |