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Preface

I am interested in the following familiy of $n×n$ matrices (empty entries are zeros):

$$ A = \begin{pmatrix} \newcommand{\h}{\hphantom{-}} 1 & \h 1 & \h 1 & \cdots & \h 1 \\ 1 & -1 & & & \\ & \h 1 & -1 & & \\ & & \ddots & \ddots & \\ & & & \h 1 & -1 \\ \end{pmatrix}$$

Such matrices arise when you take the vector of ones (first row) and try to systematically expand it to a basis using only sparse vectors orthogonal to it but not necessarily to each other (the other rows). This problem in turn comes up when trying to transform a dynamical system such that the synchronisation manifold is represented by a single component (the vector of ones).

Empirically, it is easy to see that the inverses of these matrices are:

$$ \newcommand{\n}{\hphantom{n}} \newcommand{\mn}{\hphantom{-n}} A^{-1}= \frac{1}{n}\; \begin{pmatrix} 1 & n-1 & n-2 & n-3 & \cdots & 1 \\ 1 & \n -1 & n-2 & n-3 & \cdots & 1 \\ 1 & \n -1 & \n -2 & n-3 & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ 1 & \n -1 & \n -2 & \cdots & 2-n & 1 \\ 1 & \n -1 & \n -2 & \cdots & 2-n & 1-n \\ \end{pmatrix} $$ I am pretty confident that I can prove this if I must (if not I will ask a separate question).

Actual Question

Do these matrices have a name? Is there any literature about them and preferably a proof of the inverse? I am asking to be able to avoid a lengthy and tedious elaboration on the subject in a paper and instead be able to just drop a name and a reference.

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