Given a matrix $Ainmathbb{C}^{n imes n}$ 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 Jordan canonical form (JCF) of $A$. Assume that $A$ is nonsingular and that $V$ and $J$ are given. How to obtain $widehat{V}$ and $widehat{J}$ such that $widehat{V}^{-1}A^{-1}widehat{V}=widehat{J}$ and $widehat{J}$ 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 companion 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

Enrico Bozzo;
In corso di stampa

Abstract

Given a matrix $Ainmathbb{C}^{n imes n}$ 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 Jordan canonical form (JCF) of $A$. Assume that $A$ is nonsingular and that $V$ and $J$ are given. How to obtain $widehat{V}$ and $widehat{J}$ such that $widehat{V}^{-1}A^{-1}widehat{V}=widehat{J}$ and $widehat{J}$ 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 companion are non-derogatory lower Hessenberg matrices. The answers to the two questions will be obtained by solving two linear matrix equations involving these matrices.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11390/1224730.6
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