The relationship between the eigenvalues and eigenvectors of a similarity matrix and its associated Carbo index matrix

作者：Ayers Paul W^*; Carbo Dorca Ramon

来源：Journal of Mathematical Chemistry, 2011, 49(1): 6-11.

DOI：10.1007/s10910-010-9737-8

• Quantum similarity
• Carbo index
• Generalized eigenvalue problem

The eigenvalues and eigenvectors of a quantum similarity matrix are also generalized eigenvalues and eigenvectors of the associated matrix of Carbo indices. This establishes bounds on the spectrum of the Carbo index matrix; for example, a quantum similarity matrix is positive semidefinite if and only if the associated Carbo index matrix is also positive semidefinite. The generalized eigenvalue problem for the Carbo index matrix has a diagonal metric matrix on the right-hand-side. Every generalized eigenvalue problem can be written in this diagonal form (i.e., this form is not special to this application). This diagonally structure generalized eigenvalue problem is especially convenient because it can be converted to a conventional eigenvalue problem by a particularly simple partial Lowdin transformation.

• 出版日期2011-1