Arrowhead matrix

In themathematicalfield oflinear algebra,anarrowhead matrixis asquare matrixcontaining 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

$A={\begin{bmatrix}\,\!*&*&*&*&*\\\,\!*&*&0&0&0\\\,\!*&0&*&0&0\\\,\!*&0&0&*&0\\\,\!*&0&0&0&*\end{bmatrix}}.$

Any symmetric permutation of the arrowhead matrix,$P^{T}AP$,wherePis apermutation matrix,is a(permuted) arrowhead matrix.Real symmetric arrowhead matrices are used in some algorithms for finding ofeigenvaluesandeigenvectors.

Real symmetric arrowhead matrices

LetAbe a real symmetric (permuted) arrowhead matrix of the form

$A={\begin{bmatrix}D&z\\z^{T}&\alpha \end{bmatrix}},$

where$D=\mathop {\mathrm {diag} } (d_{1},d_{2},\ldots,d_{n-1})$is diagonal matrix of ordern−1,$z={\begin{bmatrix}\zeta _{1}&\zeta _{2}&\cdots &\zeta _{n-1}\end{bmatrix}}^{T}$is a vector and$\alpha$is a scalar. Note that here the arrow is pointing to the bottom right.

Let

$A=V\Lambda V^{T}$

be theeigenvalue decompositionofA,where $\Lambda =\operatorname {diag} (\lambda _{1},\lambda _{2},\ldots,\lambda _{n})$ is a diagonal matrix whose diagonal elements are theeigenvaluesofA,and $V={\begin{bmatrix}v_{1}&\cdots &v_{n}\end{bmatrix}}$ is an orthonormal matrix whose columns are the correspondingeigenvectors.The following holds:

• If$\zeta _{i}=0$for somei,then the pair$(d_{i},e_{i})$,where$e_{i}$is thei-thstandard basisvector, is an eigenpair ofA.Thus, all such rows and columns can be deleted, leaving the matrix with all$\zeta _{i}\neq 0$.
• TheCauchy interlacing theoremimplies that the sorted eigenvalues ofAinterlace the sorted elements$d_{i}$:if$d_{1}\geq d_{2}\geq \cdots \geq d_{n-1}$(this can be attained by symmetric permutation of rows and columns without loss of generality), and if$\lambda _{i}$s are sorted accordingly, then$\lambda _{1}\geq d_{1}\geq \lambda _{2}\geq d_{2}\geq \cdots \geq \lambda _{n-1}\geq d_{n-1}\geq \lambda _{n}$.
• If$d_{i}=d_{j}$,for some$i\neq j$,the above inequality implies that$d_{i}$is an eigenvalue ofA.The size of the problem can be reduced by annihilating$\zeta _{j}$with aGivens rotationin the$(i,j)$-plane and proceeding as above.

Symmetric arrowhead matrices arise in descriptions of radiationlesstransitionsin isolated molecules and oscillators vibrationally coupled with aFermi liquid.

Eigenvalues and eigenvectors

A symmetric arrowhead matrix isirreducibleif$\zeta _{i}\neq 0$for alliand$d_{i}\neq d_{j}$for all$i\neq j$.Theeigenvaluesof an irreducible real symmetric arrowhead matrix are the zeros of thesecular equation

$f(\lambda )=\alpha -\lambda -\sum _{i=1}^{n-1}{\frac {\zeta _{i}^{2}}{d_{i}-\lambda }}\equiv \alpha -\lambda -z^{T}(D-\lambda I)^{-1}z=0$

which can be, for example, computed by thebisection method.The correspondingeigenvectorsare equal to

$v_{i}={\frac {x_{i}}{\|x_{i}\|_{2}}},\quad x_{i}={\begin{bmatrix}\left(D-\lambda _{i}I\right)^{-1}z\\-1\end{bmatrix}},\quad i=1,\ldots,n.$

Direct application of the above formula may yield eigenvectors which are not numerically sufficiently orthogonal. The forward stable algorithm which computes each eigenvalue and each component of the corresponding eigenvector to almost full accuracy is described in.TheJuliaversion of the software is available.

Inverses

LetAbe an irreducible real symmetric (permuted) arrowhead matrix of the form

$A={\begin{bmatrix}D&z\\z^{T}&\alpha \end{bmatrix}}.$

If$d_{i}\neq 0$for alli,the inverse is arank-one modification of a diagonal matrix(diagonal-plus-rank-onematrix orDPR1):

$A^{-1}={\begin{bmatrix}D^{-1}&\\&0\end{bmatrix}}+\rho uu^{T},$

where

$u={\begin{bmatrix}D^{-1}z\\-1\end{bmatrix}},\quad \rho ={\frac {1}{\alpha -z^{T}D^{-1}z}}.$

If$d_{i}=0$for somei,the inverse is a permuted irreducible real symmetric arrowhead matrix:

$A^{-1}={\begin{bmatrix}D_{1}^{-1}&w_{1}&0&0\\w_{1}^{T}&b&w_{2}^{T}&1/\zeta _{i}\\0&w_{2}&D_{2}^{-1}&0\\0&1/\zeta _{i}&0&0\end{bmatrix}}$

where

${\begin{alignedat}{2}D_{1}&=\mathop {\mathrm {diag} } (d_{1},d_{2},\ldots,d_{i-1}),\\D_{2}&=\mathop {\mathrm {diag} } (d_{i+1},d_{i+2},\ldots,d_{n-1}),\\z_{1}&={\begin{bmatrix}\zeta _{1}&\zeta _{2}&\cdots &\zeta _{i-1}\end{bmatrix}}^{T},\\z_{2}&={\begin{bmatrix}\zeta _{i+1}&\zeta _{i+2}&\cdots &\zeta _{n-1}\end{bmatrix}}^{T},\\w_{1}&=-D_{1}^{-1}z_{1}{\frac {1}{\zeta _{i}}},\\w_{2}&=-D_{2}^{-1}z_{2}{\frac {1}{\zeta _{i}}},\\b&={\frac {1}{\zeta _{i}^{2}}}\left(-a+z_{1}^{T}D_{1}^{-1}z_{1}+z_{2}^{T}D_{2}^{-1}z_{2}\right).\end{alignedat}}$