Abstract
A novel relation between the Moore-Penrose inverses of two nullity-1 $n\times n$ Hermitian matrices which share a common null eigenvector is established, and its application in electrical networks is illustrated by applying the result to Laplacian matrices of graphs.
Highlights
The Hermitian matrices are an important class of matrices arising in many contexts
A complex squared matrix is called a Hermitian matrix if it is equal to its conjugate transpose, in other words, for all i and j, its (i, j)-th element is equal to the complex conjugate of its (j, i)-th element
A Relation Between Moore-Penrose that the resistance distance turns out to have many purely mathematical interpretations, it comes from physics and engineering, among which a fundamental one is the classical result which is given via the Moore-Penrose inverse of the Laplacian matrix [2]: (i, j) = L+ii − 2L+ij + L+jj, (1.1)
Summary
The Hermitian matrices are an important class of matrices arising in many contexts. A complex squared matrix is called a Hermitian matrix if it is equal to its conjugate transpose, in other words, for all i and j, its (i, j)-th element (i.e., the element in the i-th row and j-th column) is equal to the complex conjugate of its (j, i)-th element. It is natural to consider a weighted graph G as a (resistive) electrical network N by viewing each edge e as a resistor such that the conductance of the resistor is we, where we is the weight on e In this guise, the resistance distance [2] between any two vertices i and j of G, denoted by (i, j), is defined as the net effective resistance between corresponding nodes i and j in N. A Relation Between Moore-Penrose that the resistance distance turns out to have many purely mathematical interpretations, it comes from physics and engineering, among which a fundamental one is the classical result which is given via the Moore-Penrose inverse of the Laplacian matrix [2]:. Its application in electrical networks is illustrated by applying the result to Laplacian matrices of graphs
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