Arrowhead matrix
In the mathematical field of linear algebra, an arrowhead matrix is a square matrix containing zeros in all entries except for the first row, first column, and main diagonal, these entries can be any number. In other words, the matrix has the form Any symmetric permutation of the arrowhead matrix, , where P is a permutation matrix, is a (permuted) arrowhead matrix. Real symmetric arrowhead matrices are used in some algorithms for finding of eigenvalues and eigenvectors. Real symmetric arrowhead matrices Let A be a real symmetric (permuted) arrowhead matrix of the form where is diagonal matrix of order n−1, is a vector and is a scalar. Let be the eigenvalue decomposition of A, where is a diagonal matrix whose diagonal elements are the eigenvalues of A, and is an orthonormal matrix whose columns are the corresponding eigenvectors. The following holds: If for some i, then the pair , where is the i-th standard basis vector, is an eigenpair of A. Thus, all such rows and columns can be deleted, leaving the matrix with all . The Cauchy interlacing theorem implies that the sorted eigenvalues of A interlace the sorted elements : if (this can be attained by symmetric permutation