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A note on the singular value decomposition of idempotent and involutory matrices

Heike Faßbender, Martin Halwaß

TL;DR

The paper characterizes the singular-value structure of idempotent and involutory matrices by deriving explicit SVDs and exact counts of singular values: for an idempotent $M$ of rank $r$, there are $n-r$ zeros, $t=\\operatorname{rank}(M)-\\operatorname{dim}(\\operatorname{null}(I-MM^H))$ singular values with $\\sigma>1$, and $r-t$ with $\\sigma=1$; for the involutory $B=2M-I$, singular values appear in reciprocal pairs with a subset equal to one, and the non-unit values relate to the idempotent case via $\\sigma_j=\\tan\\phi_j$, where $\\phi_j=\\tfrac12(\\tfrac{\\pi}{2}+\\psi_j)$. The results leverage a condensed Schur form and principal-angle data to reveal a tight correspondence between left and right singular vectors, providing explicit SVDs and illuminating the spectral geometry of these matrices.

Abstract

It is known that singular values of idempotent matrices are either zero or larger or equal to one \cite{HouC63}. We state exactly how many singular values greater than one, equal to one, and equal to zero there are. Moreover, we derive a singular value decomposition of idempotent matrices which reveals a tight relationship between its left and right singular vectors. The same idea is used to augment a discovery regarding the singular values of involutory matrices as presented in \cite{FasH20}.

A note on the singular value decomposition of idempotent and involutory matrices

TL;DR

The paper characterizes the singular-value structure of idempotent and involutory matrices by deriving explicit SVDs and exact counts of singular values: for an idempotent of rank , there are zeros, singular values with , and with ; for the involutory , singular values appear in reciprocal pairs with a subset equal to one, and the non-unit values relate to the idempotent case via , where . The results leverage a condensed Schur form and principal-angle data to reveal a tight correspondence between left and right singular vectors, providing explicit SVDs and illuminating the spectral geometry of these matrices.

Abstract

It is known that singular values of idempotent matrices are either zero or larger or equal to one \cite{HouC63}. We state exactly how many singular values greater than one, equal to one, and equal to zero there are. Moreover, we derive a singular value decomposition of idempotent matrices which reveals a tight relationship between its left and right singular vectors. The same idea is used to augment a discovery regarding the singular values of involutory matrices as presented in \cite{FasH20}.
Paper Structure (3 sections, 3 theorems, 35 equations)

This paper contains 3 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Singular value decomposition of idempotent matrices
  3. An SVD of involutory matrices

Key Result

Lemma 1

Let $M\in \mathbb{C}^{n \times n}$ be an idempotent matrix of rank $r$ with the SVD $M = U \Sigma V^H.$ Let $u_j$ and $v_j$ be the left and right singular vectors, respectively, of $M$ associated with the nonzero singular values $\sigma_j, j = 1, 2, \ldots, r.$ Then, $u_j^Hv_j = \sigma_j^{-1}.$

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2: SVD of $N$
  • Proof 1
  • Theorem 3: SVD of idempotent matrices
  • Remark 4
  • Remark 5