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

#### 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.
##### Scheda breve Scheda completa Scheda completa (DC)
File in questo prodotto:
File
perEnrico.pdf

non disponibili

Descrizione: Articolo principale
Tipologia: Documento in Post-print
Licenza: Non pubblico
Dimensione 250.18 kB
LAMACARMINE.pdf

non disponibili

Descrizione: Articolo principale
Tipologia: Versione Editoriale (PDF)
Licenza: Non pubblico
Dimensione 510.85 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11390/1224730.6`
• ND
• ND
• ND