Given a matrix A is an element of C-nxn there exists a nonsingular matrix V such that V-1 AV = J, where J is a very sparse matrix with a diagonal block structure, known as the Jordan canonical form (JCF) of A. Assume that A is nonsingular and that V and J are given. How to obtain (V) over cap and (J) over cap such that (V) over cap (-1) A(-1) (V) over cap = (J) over cap and (J) over cap is the JCF of A(-1) ? Curiously, the answer involves the Pascal matrix. For the Frobenius canonical form (FCF), where blocks are companion matrices, the analogous question has a very simple answer. Jordan blocks and companions are non-derogatory lower Hessenberg matrices. The answers to the two questions will be obtained by solving two linear matrix equations involving these matrices.
The Jordan and Frobenius pairs of the inverse
Deidda, Piero;
2022
Abstract
Given a matrix A is an element of C-nxn there exists a nonsingular matrix V such that V-1 AV = J, where J is a very sparse matrix with a diagonal block structure, known as the Jordan canonical form (JCF) of A. Assume that A is nonsingular and that V and J are given. How to obtain (V) over cap and (J) over cap such that (V) over cap (-1) A(-1) (V) over cap = (J) over cap and (J) over cap is the JCF of A(-1) ? Curiously, the answer involves the Pascal matrix. For the Frobenius canonical form (FCF), where blocks are companion matrices, the analogous question has a very simple answer. Jordan blocks and companions are non-derogatory lower Hessenberg matrices. The answers to the two questions will be obtained by solving two linear matrix equations involving these matrices.Pubblicazioni consigliate
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