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Generalized Eigenvector and Transition Matrix

Yongheng Chen

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Abstract

The generalized eigenvector of the matrix A and the A's eigenvector in use were described.The equation,(A-λE)x=ζ in application was used to find A's high rank of generalized eigenvector gradually from the low rank of generalized eigenvector.It was proved for the first time that the generalized eigenvector by this method,the matrix A had nothing to do with linearity.Meanwhile,n-order matrix just had n pieces of the independent generalized eigenvectors.The application of these generalized eigenvectors to making transition matrix P,and having P-1AP be Jordan canonical matrix was proposed.

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What this paper is about

The generalized eigenvector of the matrix A and the A's eigenvector in use were described.The equation,(A-λE)x=ζ in application was used to find A's high rank of generalized eigenvector gradually from the low rank of generalized eigenvector.It was proved for the first time that the generalized eigenvector by this method,the matrix A had nothing to do with linearity.Meanwhile,n-order matrix just had n pieces of the independent generalized eigenvectors.The application of these generalized eigenvectors to making transition matrix P,and having P-1AP be Jordan canonical matrix was proposed.

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Available abstract

The generalized eigenvector of the matrix A and the A's eigenvector in use were described.The equation,(A-λE)x=ζ in application was used to find A's high rank of generalized eigenvector gradually from the low rank of generalized eigenvector.It was proved for the first time that the generalized eigenvector by this method,the matrix A had nothing to do with linearity.Meanwhile,n-order matrix just had n pieces of the independent generalized eigenvectors.The application of these generalized eigenvectors to making transition matrix P,and having P-1AP be Jordan canonical matrix was proposed.

Key concepts: Generalized eigenvector, Eigenvalues and eigenvectors, Defective matrix, Mathematics, Matrix (chemical analysis), Rank (graph theory), Matrix differential equation, Eigendecomposition of a matrix

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