The generalized singular value decomposition and the general (A-λB)-problem

In the study of the canonical structure of matrix pencilsA-λB, the column and row nullities ofA andB and the possible common nullspaces give information about the Kronecker structure ofA-λB. It is shown how to extract the significant information concerning these nullspaces from the generalized singular value decomposition (GSVD) of the matrix pair (A, B), These properties are the basis for the RGSVD-algorithm that will be published elsewhere [9]. An algorithm for the numerical computation of the transformation matrices (V, X), used in an equivalence transformationVH(A-λB)X, is presented and discussed. A generalizedQZ-decomposition (GQZD) of a matrix pair (A, B) is also formulated, giving a unitary equivalent transformationVH(A-λB)Q. If during the computationsX gets too illconditioned we switch to the GQZD, sacrificing a simple diagonal structure of the transformedB-part in order to maintain stability.

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