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 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, P T A P {displaystyle P^{T}AP} , 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. Let A be a real symmetric (permuted) arrowhead matrix of the form where D = d i a g ⁡ ( d 1 , d 2 , … , d n − 1 ) {displaystyle D=mathop {mathrm {diag} } (d_{1},d_{2},ldots ,d_{n-1})} is diagonal matrix of order n-1, z = [ ζ 1 ζ 2 ⋯ ζ n − 1 ] T {displaystyle z={egin{bmatrix}zeta _{1}&zeta _{2}&cdots &zeta _{n-1}end{bmatrix}}^{T}} is a vector and α {displaystyle alpha } is a scalar. Let be the eigenvalue decomposition of A, where Λ = d i a g ⁡ ( λ 1 , λ 2 , … , λ n ) {displaystyle Lambda =mathop {mathrm {diag} } (lambda _{1},lambda _{2},ldots ,lambda _{n})} is a diagonal matrix whose diagonal elements are the eigenvalues of A, and V = [ v 1 ⋯ v n ] {displaystyle V={egin{bmatrix}v_{1}&cdots &v_{n}end{bmatrix}}} is an orthonormal matrix whose columns are the corresponding eigenvectors. The following holds: Symmetric arrowhead matrices arise in descriptions of radiationless transitions in isolated molecules and oscillators vibrationally coupled with a Fermi liquid.