![Ch 3](http://s1.studyres.com/store/data/018121613_1-4c6ba3788a5c5881c7c2fede4ea174fb-300x300.png)
Spectral properties of the hierarchical product of graphs
... of the hierarchical product as a function of the eigenvalues and eigenvectors of the two underlying networks and a coupling parameter that tunes their interactions. Due to the central role of the Perron-Frobenius (PF) eigenvalue [20], i.e., the largest eigenvalue, of the adjacency matrix in several ...
... of the hierarchical product as a function of the eigenvalues and eigenvectors of the two underlying networks and a coupling parameter that tunes their interactions. Due to the central role of the Perron-Frobenius (PF) eigenvalue [20], i.e., the largest eigenvalue, of the adjacency matrix in several ...
Sec 3 Add Maths : Matrices
... gives the number of possible two-stage routes from place to place. E.g. the number in the 1st row, 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA, another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road. ...
... gives the number of possible two-stage routes from place to place. E.g. the number in the 1st row, 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA, another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road. ...
Reading Assignment 6
... A matrix is a rectangular array of numbers, parameters, or variables placed in rows and columns. The elements (members) of the matrix are usually enclosed in brackets. Ordering of the elements into rows and columns is important. The number of rows and the number of columns in a matrix together defin ...
... A matrix is a rectangular array of numbers, parameters, or variables placed in rows and columns. The elements (members) of the matrix are usually enclosed in brackets. Ordering of the elements into rows and columns is important. The number of rows and the number of columns in a matrix together defin ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.