Arrowhead matrix

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

:$$

A = \begin{bmatrix}

D & z \\

z^{T} & \alpha

\end{bmatrix},

where $D=\backslash mathop\{\backslash mathrm\{diag\}\}(d\_\{1\},d\_\{2\},\backslash ldots\; ,d\_\{n-1\})$ is diagonal matrix of order n−1, $z=\backslash begin\{bmatrix\}$

\zeta _{1} & \zeta _{2} & \cdots & \zeta _{n-1}

\end{bmatrix}^T is a vector and $\backslash alpha$ 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

$\backslash Lambda\; =\; \backslash operatorname\{diag\}(\backslash lambda\_1,\backslash lambda\_2,\backslash ldots\; ,\backslash lambda\_n)$

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 $\backslash zeta\_i=0$ 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 $\backslash zeta\_i\backslash neq\; 0$.
• The Cauchy interlacing theorem implies that the sorted eigenvalues of A interlace the sorted elements $d\_i$: if $d\_1\; \backslash geq\; d\_2\backslash geq\; \backslash cdots\backslash geq\; d\_\{n-1\}$ (this can be attained by symmetric permutation of rows and columns without loss of generality), and if $\backslash lambda\_i$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 $i\backslash neq\; j$, the above inequality implies that $d\_\{i\}$ is an eigenvalue of A. The size of the problem can be reduced by annihilating $\backslash zeta\_j$ 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.{{cite journal|last1=O'Leary|first1=D.P.|last2=Stewart|first2=G.W.|title=Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices|journal=Journal of Computational Physics|date=October 1990|volume=90|issue=2|pages=497–505|doi=10.1016/0021-9991(90)90177-3|bibcode=1990JCoPh..90..497O|url=https://zenodo.org/record/1253916}}

A symmetric arrowhead matrix is irreducible if $\backslash zeta\_i\backslash neq\; 0$ for all i and $d\_\{i\}\backslash neq\; d\_\{j\}$ for all $i\backslash neq\; j$. 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.

The forward stable algorithm which computes each eigenvalue and each component of the corresponding eigenvector to almost full accuracy is described in. The Julia version of the software is available.[https://github.com/ivanslapnicar/Arrowhead.jl "Arrowhead.jl"]

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

:$$

A = \begin{bmatrix}

D & z \\

z^{T} & \alpha

\end{bmatrix}.

If $d\_i\backslash neq\; 0$ 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{alignat}{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{alignat}

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Category:Matrices (mathematics)