[Paper Review] A Ring Isomorphism and corresponding Pseudoinverses

Samuel N. Cohen, Robert J. Elliott+1 more|ArXiv.org|Oct 1, 2008

Matrix Theory and Algorithms4 references3 citations

TL;DR

This paper establishes a ring isomorphism between the set of $n\times n$ matrices with zero row and column sums and the full set of $(n-1)\times(n-1)$ matrices, using a transformation via a matrix $J_n$ that embeds the lower-dimensional space. The isomorphism preserves the Moore-Penrose pseudoinverse structure, mapping it to the standard inverse in the reduced space, enabling explicit computation of pseudoinverses and characterization of the projection operator $X^+X$ for rank-deficient matrices in this class.

ABSTRACT

This paper studies the set of $n imes n$ matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and furthermore is isomorphic to the set of arbitrary $(n-1) imes (n-1)$ matrices. The Moore-Penrose pseudoinverse corresponds with the true inverse, (when it exists), in this lower dimension and an explicit representation of this pseudoinverse in terms of the lower dimensional space is given. This analysis is then extended to non-square matrices with all row or all column sums equal to zero.

Motivation & Objective

• To analyze the algebraic structure of $n\times n$ matrices with all row and column sums equal to zero.
• To resolve the issue that such matrices are singular (due to the all-ones vector as a null eigenvector), preventing standard matrix inversion.
• To develop a systematic method for computing Moore-Penrose pseudoinverses for these singular matrices.
• To characterize the range of the projection operator $X^+X$ for such matrices, particularly in the rank-deficient case.
• To extend the framework to non-square matrices and matrices with zero rows/columns, preserving the isomorphism and pseudoinverse correspondence.

Proposed method

• Define a transformation $\phi: M_n \to S_{n+1}$ via $\phi(X) = J_n^* X J_n$, where $J_n = [I_n \mid -\mathbf{1}]$, mapping $n\times n$ matrices to $(n+1)\times(n+1)$ matrices with zero row and column sums.
• Introduce a twisted product $X \circ Y = X K_n Y$, where $K_n = J_n J_n^*$, to define a ring structure on $M_n$ isomorphic to $S_{n+1}$ under standard matrix operations.
• Prove that $\phi$ is a ring isomorphism, preserving addition and multiplication, and that $\phi$ maps the Moore-Penrose pseudoinverse in $M_n$ under $\circ$ to the Moore-Penrose pseudoinverse in $S_{n+1}$ under standard multiplication.
• Extend the isomorphism to matrices with zero rows and columns by introducing modified $J_{m,\mathbf{a}}$ matrices that insert zero columns at specified positions $\mathbf{a}$, preserving the isomorphism and $K_m = J_{m,\mathbf{a}} J_{m,\mathbf{a}}^*$.
• Derive an explicit formula for the Moore-Penrose inverse of $\tilde{X} = J_{m,\mathbf{b}}^* X J_{m,\mathbf{a}}$ as $\tilde{X}^+ = J_{m,\mathbf{a}}^* K_m^{-1} X^{-1} K_m^{-1} J_{m,\mathbf{b}}$ when $X$ is invertible.
• Characterize the range of $\tilde{X}^+ \tilde{X}$ as the set of matrices $\tilde{M} = J_{m,\mathbf{a}}^* M$ with zero column sums and zeros in positions indexed by $\mathbf{a}$, showing $\tilde{X}^+ \tilde{X} \tilde{M} = \tilde{M}$.

Experimental results

Research questions

• RQ1 Can the set of $n\times n$ matrices with zero row and column sums be endowed with a ring structure isomorphic to that of smaller-dimensional matrices?
• RQ2 How does the Moore-Penrose pseudoinverse of such singular matrices relate to the inverse in the isomorphic lower-dimensional ring?
• RQ3 What is the explicit form of the Moore-Penrose pseudoinverse for matrices in $S_{n+1}$ with zero rows and columns?
• RQ4 How can the range of the projection operator $X^+X$ be characterized for these matrices?
• RQ5 Can the isomorphism and pseudoinverse correspondence be extended to non-square matrices with zero row or column sums?

Key findings

• The set $S_{n+1}$ of $n\times n$ matrices with all row and column sums zero is isomorphic to the full matrix algebra $M_n$ under a twisted product $X \circ Y = X K_n Y$, with $K_n = J_n J_n^*$.
• The isomorphism $\phi(X) = J_n^* X J_n$ maps the Moore-Penrose pseudoinverse in the $M_n$ ring under $\circ$ to the Moore-Penrose pseudoinverse in $S_{n+1}$ under standard multiplication.
• For a matrix $\tilde{X} = J_{m,\mathbf{b}}^* X J_{m,\mathbf{a}}$ of rank $m$, the Moore-Penrose inverse is explicitly given by $\tilde{X}^+ = J_{m,\mathbf{a}}^* K_m^{-1} X^{-1} K_m^{-1} J_{m,\mathbf{b}}$, provided $X$ is invertible.
• The projection operator $\tilde{X}^+ \tilde{X}$ acts as the identity on the subspace of matrices $\tilde{M} = J_{m,\mathbf{a}}^* M$, which are those with zero column sums and zeros in the rows indexed by $\mathbf{a}$, and $\tilde{X}^+ \tilde{X} \tilde{M} = \tilde{M}$.
• The range of $\tilde{X}^+ \tilde{X}$ is the $m$-dimensional subspace of matrices $\tilde{M} = J_{m,\mathbf{a}}^* M$, and $\tilde{X}^+ \tilde{X} = J_{m,\mathbf{a}}^* K_m^{-1} J_{m,\mathbf{a}} = J_{m,\mathbf{a}}^* \left(I_m - \frac{1}{m+1} \mathbf{1}_{m\times m}\right) J_{m,\mathbf{a}}$.
• The isomorphism and pseudoinverse correspondence extend to non-square matrices by allowing different zero insertion patterns $\mathbf{a}$ and $\mathbf{b}$ in the $J_{m,\mathbf{a}}$ and $J_{m,\mathbf{b}}$ matrices, preserving the structure and invertibility conditions.