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.^[1][2] In other words, the matrix has the form

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

Any symmetric permutation of the arrowhead matrix, ${\displaystyle P^{T}AP}$, 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.^[3]

Real symmetric arrowhead matrices

Let A be a real symmetric (permuted) arrowhead matrix of the form

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

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

Let

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

be the eigenvalue decomposition of A, where ${\displaystyle \Lambda =\operatorname {diag} (\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})}$ is a diagonal matrix whose diagonal elements are the eigenvalues of A, and ${\displaystyle V={\begin{bmatrix}v_{1}&\cdots &v_{n}\end{bmatrix}}}$ is an orthonormal matrix whose columns are the corresponding eigenvectors. The following holds:

• If ${\displaystyle \zeta _{i}=0}$ for some i, then the pair ${\displaystyle (d_{i},e_{i})}$, where ${\displaystyle e_{i}}$ 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 ${\displaystyle \zeta _{i}\neq 0}$.
• The Cauchy interlacing theorem implies that the sorted eigenvalues of A interlace the sorted elements ${\displaystyle d_{i}}$: if ${\displaystyle d_{1}\geq d_{2}\geq \cdots \geq d_{n-1}}$ (this can be attained by symmetric permutation of rows and columns without loss of generality), and if ${\displaystyle \lambda _{i}}$s are sorted accordingly, then ${\displaystyle \lambda _{1}\geq d_{1}\geq \lambda _{2}\geq d_{2}\geq \cdots \geq \lambda _{n-1}\geq d_{n-1}\geq \lambda _{n}}$.
• If ${\displaystyle d_{i}=d_{j}}$, for some ${\displaystyle i\neq j}$, the above inequality implies that ${\displaystyle d_{i}}$ is an eigenvalue of A. The size of the problem can be reduced by annihilating ${\displaystyle \zeta _{j}}$ with a Givens rotation in the ${\displaystyle (i,j)}$-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 ${\displaystyle \zeta _{i}\neq 0}$ for all i and ${\displaystyle d_{i}\neq d_{j}}$ for all ${\displaystyle i\neq j}$. The eigenvalues of an irreducible real symmetric arrowhead matrix are the zeros of the secular equation

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

which can be, for example, computed by the bisection method. The corresponding eigenvectors are equal to

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

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

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

If ${\displaystyle d_{i}\neq 0}$ for all i, the inverse is a rank-one modification of a diagonal matrix (diagonal-plus-rank-one matrix or DPR1):

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

where

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

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

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

where

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

References