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

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

Any symmetric permutation of the arrowhead matrix, $P^{T}AP$ , 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]

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

If $\zeta _{i}=0$ for some i, then the pair $(d_{i},e_{i})$ , where $e_{i}$ 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 $\zeta _{i}\neq 0$ .
The Cauchy interlacing theorem implies that the sorted eigenvalues of A interlace 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 of A. The size of the problem can be reduced by annihilating $\zeta _{j}$ with a Givens rotation in the $(i,j)$ -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 $\zeta _{i}\neq 0$ for all i and $d_{i}\neq d_{j}$ for all $i\neq j$ . The eigenvalues of an irreducible real symmetric arrowhead matrix are the zeros of the secular 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 the bisection method. The corresponding eigenvectors are 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.^[1] The forward stable algorithm which computes each eigenvalue and each component of the corresponding eigenvector to almost full accuracy is described in.^[2]

Inverses

Let A be 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 all i, the inverse is a rank-one modification of a diagonal matrix (diagonal-plus-rank-one matrix or DPR1):

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 some i, 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}}

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Eigenvalues and eigenvectors

Inverses

References

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