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I am considering a graph with $n$ edges with the following nicely structured adjacency matrix: \begin{equation} A_n= \begin{pmatrix} 0 & 0 & 0 &\cdots & 0 & 0 & 1\\ 0 & 0 & 0 &\cdots & 0 & 1 & 1\\ \vdots & \vdots & & & & \vdots & \vdots\\ 0 & 1 & 1 &\cdots & 1 & 1 & 1\\ 1 & 1 & 1 &\cdots & 1 & 1 & 1\\ \end{pmatrix}. \end{equation} I need to determine the pseudoinverse $L_n^+$ of its Laplacian: \begin{equation} L_n=\text{diag}(1, ...,n)-A. \end{equation}\begin{equation} L_n=\text{diag}(1, ...,n)-A_n. \end{equation} After playing around with Mathematica, $L_n^+$ seems to have a nice structure. However, I am not so familiar with determining pseudoinverses and therefore would like some help in determining this structure.

My question is, how would one go about calculating pseudoinverses and is it possible to get a closed-form expression for $L_n^+$ for general $n$? Thanks in advance.

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix: \begin{equation} A_n= \begin{pmatrix} 0 & 0 & 0 &\cdots & 0 & 0 & 1\\ 0 & 0 & 0 &\cdots & 0 & 1 & 1\\ \vdots & \vdots & & & & \vdots & \vdots\\ 0 & 1 & 1 &\cdots & 1 & 1 & 1\\ 1 & 1 & 1 &\cdots & 1 & 1 & 1\\ \end{pmatrix}. \end{equation} I need to determine the pseudoinverse $L_n^+$ of its Laplacian: \begin{equation} L_n=\text{diag}(1, ...,n)-A. \end{equation} After playing around with Mathematica, $L_n^+$ seems to have a nice structure. However, I am not so familiar with determining pseudoinverses and therefore would like some help in determining this structure.

My question is, how would one go about calculating pseudoinverses and is it possible to get a closed-form expression for $L_n^+$ for general $n$? Thanks in advance.

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix: \begin{equation} A_n= \begin{pmatrix} 0 & 0 & 0 &\cdots & 0 & 0 & 1\\ 0 & 0 & 0 &\cdots & 0 & 1 & 1\\ \vdots & \vdots & & & & \vdots & \vdots\\ 0 & 1 & 1 &\cdots & 1 & 1 & 1\\ 1 & 1 & 1 &\cdots & 1 & 1 & 1\\ \end{pmatrix}. \end{equation} I need to determine the pseudoinverse $L_n^+$ of its Laplacian: \begin{equation} L_n=\text{diag}(1, ...,n)-A_n. \end{equation} After playing around with Mathematica, $L_n^+$ seems to have a nice structure. However, I am not so familiar with determining pseudoinverses and therefore would like some help in determining this structure.

My question is, how would one go about calculating pseudoinverses and is it possible to get a closed-form expression for $L_n^+$ for general $n$? Thanks in advance.

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MthQ
MthQ
  • 41
  • 3
  • 15

Calculating a generalized inverse (Moore–Penrose pseudoinverse)

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix: \begin{equation} A_n= \begin{pmatrix} 0 & 0 & 0 &\cdots & 0 & 0 & 1\\ 0 & 0 & 0 &\cdots & 0 & 1 & 1\\ \vdots & \vdots & & & & \vdots & \vdots\\ 0 & 1 & 1 &\cdots & 1 & 1 & 1\\ 1 & 1 & 1 &\cdots & 1 & 1 & 1\\ \end{pmatrix}. \end{equation} I need to determine the pseudoinverse $L_n^+$ of its Laplacian: \begin{equation} L_n=\text{diag}(1, ...,n)-A. \end{equation} After playing around with Mathematica, $L_n^+$ seems to have a nice structure. However, I am not so familiar with determining pseudoinverses and therefore would like some help in determining this structure.

My question is, how would one go about calculating pseudoinverses and is it possible to get a closed-form expression for $L_n^+$ for general $n$? Thanks in advance.