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.[1][2] In other words, the matrix has the form
![{\displaystyle A={\begin{bmatrix}\,\!*&*&*&*&*\\\,\!*&*&0&0&0\\\,\!*&0&*&0&0\\\,\!*&0&0&*&0\\\,\!*&0&0&0&*\end{bmatrix)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3774b151a5ec2e68d00e77a43da15f18b39a58c4)
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.[3]
Real symmetric arrowhead matrices
Let A be a real symmetric (permuted) arrowhead matrix of the form
![{\displaystyle A={\begin{bmatrix}D&z\\z^{T}&\alpha \end{bmatrix)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28184a9d99e18ed7050dff73879e955c9b8ab086)
where
is diagonal matrix of order n−1,
is a vector and
is a scalar. Note that here the arrow is pointing to the bottom right.
Let
![{\displaystyle A=V\Lambda V^{T))](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d647f5a1d8743def9bb8f51af4894860a3b30e)
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 of rows and columns without loss of generality), and if
s are sorted accordingly, then
.
- If
, for some
, the above inequality implies that
is an eigenvalue of A. The size of the problem can be reduced by annihilating
with a Givens rotation in the
-plane and proceeding as above.
Symmetric arrowhead matrices arise in descriptions of radiationless transitions in isolated molecules and oscillators vibrationally coupled with a Fermi liquid.[4]
Eigenvalues and eigenvectors
A symmetric arrowhead matrix is irreducible if
for all i and
for all
. The eigenvalues of an irreducible real symmetric arrowhead matrix are the zeros of the secular equation
![{\displaystyle 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37fcd7cbeafcf7c2e26c28673b03f4c22b23b383)
which can be, for example, computed by the bisection method. The corresponding eigenvectors are equal to
![{\displaystyle 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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/626c4039f4a6679e48d0accf505ce1e1dc04483c)
Direct application of the above formula may yield eigenvectors which are not numerically sufficiently orthogonal.[1]
The forward stable algorithm which computes each eigenvalue and each component of the corresponding eigenvector to almost full accuracy is described in.[2] The Julia version of the software is available.[5]
Inverses
Let A be an irreducible real symmetric (permuted) arrowhead matrix of the form
![{\displaystyle A={\begin{bmatrix}D&z\\z^{T}&\alpha \end{bmatrix)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c60188de2ab8ec8b434334da08ffa7843d564bdf)
If
for all i, the inverse is a rank-one modification of a diagonal matrix (diagonal-plus-rank-one matrix or DPR1):
![{\displaystyle A^{-1}={\begin{bmatrix}D^{-1}&\\&0\end{bmatrix))+\rho uu^{T},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22031df16bd3a306b3f12c76b264844236d97cbd)
where
![{\displaystyle u={\begin{bmatrix}D^{-1}z\\-1\end{bmatrix)),\quad \rho ={\frac {1}{\alpha -z^{T}D^{-1}z)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a047e8dad77230780c45469a990cb1e0686eae)
If
for some i, the inverse is a permuted irreducible real symmetric arrowhead matrix:
![{\displaystyle 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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d781b2ed8e6553722a4e5dddaf6692cb919e0492)
where
![{\displaystyle {\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74a80cd97b03f591dfcc8804bf23f981e870df6f)