bdqz

Block-diagonal QZ decomposition

Since R2023b

Syntax

[AA,BB,TL,TR,blks] = bdqz(A,B,condmax)

[AA,BB,TL,TR] = bdqz(A,B,[],blks)

Description

[AA,BB,TL,TR,blks] = bdqz(A,B,condmax) transforms the pair (A,B) to block-diagonal form.

The output matrices AA, BB, TL, and TR are such that $$AA=T_L^{T}A{T}_R$$ and $$BB=T_L^{T}B{T}_R$$. AA and BB are block diagonal and each diagonal pair (AAj,BBj) is in QZ form:

• AAj is quasi upper-triangular (real case) or upper-triangular (complex case).

• BBj is upper-triangular with real diagonal.

blks is a vector containing block sizes down the diagonal. Use condmax to specify an upper bound on the condition number of TL and TR. If you omit condmax, bdqz uses 1e4 as the upper bound.

[AA,BB,TL,TR] = bdqz(A,B,[],blks) specifies the desired block sizes. For this syntax, the input pair A and B must be in real or complex QZ form (see qz).

Examples

Block-Diagonal QZ Decomposition

This example shows how to perform the block-diagonal QZ decomposition.

Generate random 10-by-10 matrices with real values.

rng(0)
A = randn(10);
B = randn(10);

Perform the block-diagonal QZ decomposition.

[AA,BB,TL,TR,blks] = bdqz(A,B);
AA

AA = 10×10

    2.2815         0         0         0         0         0         0         0         0         0
         0   -1.4630         0         0         0         0         0         0         0         0
         0         0    1.0999         0         0         0         0         0         0         0
         0         0         0   -1.1764         0         0         0         0         0         0
         0         0         0         0    0.3870   -0.8050         0         0         0         0
         0         0         0         0    0.5243    0.1540         0         0         0         0
         0         0         0         0         0         0   -0.4186    0.6583         0         0
         0         0         0         0         0         0   -1.5091   -1.1076         0         0
         0         0         0         0         0         0         0         0    0.4881    0.5093
         0         0         0         0         0         0         0         0   -0.0768    0.4697

BB

BB = 10×10

    0.6842         0         0         0         0         0         0         0         0         0
         0    0.4721         0         0         0         0         0         0         0         0
         0         0    0.6384         0         0         0         0         0         0         0
         0         0         0    0.8556         0         0         0         0         0         0
         0         0         0         0    0.6615         0         0         0         0         0
         0         0         0         0         0    0.5652         0         0         0         0
         0         0         0         0         0         0    2.9601         0         0         0
         0         0         0         0         0         0         0    1.4418         0         0
         0         0         0         0         0         0         0         0    0.8740         0
         0         0         0         0         0         0         0         0         0    0.8314

blks

blks = 7×1

     1
     1
     1
     1
     2
     2
     2

bdqz returns diagonal AA and BB. The size of the blocks in this decomposition is either 1-by-1 or 2-by-2.

Input Arguments

A, B — Input matrices

matrix

Input matrices, specified as matrices with real or complex values.

Complex Number Support: Yes

condmax — Upper bound on condition number

positive scalar greater than 1

Upper bound on condition number, of TL and TR matrices, specified as a positive scalar greater than 1. If you omit this argument, bdqz uses 1e4 as the upper bound.

Use condmax to control the trade off between block size and accuracy of the decomposition. The larger condmax is, the smaller the blocks are, but TL and TR can become more ill-conditioned.

Output Arguments

AA, BB — Block-diagonal matrices

matrix

Block diagonal matrices, returned as matrices with real or complex values.

For real matrices A and B, all the diagonal blocks of AA and BB pair are real. For complex matrices A and B, each block of AA and BB is associated with a cluster of nearby eigenvalues.

For diagonalizable A and B, AA and BB are diagonal.

TL, TR — Transformation matrices

matrix

Block-diagonalizing transformation, returned as matrices.

bdqz transforms A,B to AA,BB such that $$AA=T_L^{T}A{T}_R$$ and $$BB=T_L^{T}B{T}_R$$.

blks — Block sizes

vector

Block sizes in the block-diagonal decomposition, returned as a vector.

Version History

Introduced in R2023b