Customizing subequations numbering in itemized list

Question

This is a follow-up on the question subequation numbering in itemized list.

\documentclass[10pt]{article}

% for fancy math
\usepackage{amsmath}

% rank operator
\DeclareMathOperator*{\rank}{rank}

% Matrix transpose
\newcommand{\trans}[1]{\ensuremath{{#1}^\top}}

% for extra space at the end of abbreviation
\usepackage{xspace}

% positive semi-definite
\newcommand{\psd}{\textsc{psd}\xspace}

% boldface uppercase letters for matrices
\newcommand{\Abf}{\ensuremath{\mathbf A}}
\newcommand{\Bbf}{\ensuremath{\mathbf B}}

% boldface lowercase letters for vectors
\newcommand{\xbf}{\ensuremath{\mathbf x}}

% for math blackboard font
\usepackage{amssymb}
% set of real numbers
\newcommand{\Rbb}{\ensuremath{\mathbb R}}

\usepackage{palatino}
\usepackage[sc]{mathpazo}

\begin{document}
\noindent
For any real symmetric matrcies $\Abf$ such that $\rank(\Abf_{n\times n})=r$,
the following statements are equivalent
and any one of them can serve as the definition of
\emph{positive semi-definite} (\psd) matrices.
\begin{subequations}
\begin{itemize}
  \item $\trans\xbf \Abf\xbf \geq 0$ for any non-zero vector
    $\xbf\in\Rbb^{n\times 1}$.
    \hfill\refstepcounter{equation}\textup{(\theequation)}%
  \item All the $n$ eigenvalues of $\Abf$ are non-negative.
    \hfill\refstepcounter{equation}\textup{(\theequation)}%
  \item $\Abf=\trans\Bbf \Bbf$ for some $\Bbf$ with $\rank(\Bbf)=r$.
    \hfill\refstepcounter{equation}\textup{(\theequation)}%
\end{itemize}
\end{subequations} 
\end{document}

Now, I'd like to customize the labels (1a), (1b) and (1c) to distinguish definitions from other typical equation numbers.

• I'd like to label them as (def1), (def2), and (def3).
• But I still want to use subequations environment because I want to reference them as an equation by eqref command.

How can I do this?

Answer 1

You can introduce a new counter mysub

\newcounter{mysub}
\setcounter{mysub}{0}
\renewcommand{\themysub}{def\arabic{mysub}}

and use this one instead of \theequation inside the subequations.

If you also want this counter to be reset after the end of the subequations, also add the following lines

\usepackage{etoolbox}
\AtEndEnvironment{subequations}{\setcounter{mysub}{0}}

MWE:

\documentclass[10pt]{article}

% for fancy math
\usepackage{amsmath}

% rank operator
\DeclareMathOperator*{\rank}{rank}

% Matrix transpose
\newcommand{\trans}[1]{\ensuremath{{#1}^\top}}

% for extra space at the end of abbreviation
\usepackage{xspace}

% positive semi-definite
\newcommand{\psd}{\textsc{psd}\xspace}

% boldface uppercase letters for matrices
\newcommand{\Abf}{\ensuremath{\mathbf A}}
\newcommand{\Bbf}{\ensuremath{\mathbf B}}

% boldface lowercase letters for vectors
\newcommand{\xbf}{\ensuremath{\mathbf x}}

% for math blackboard font
\usepackage{amssymb}
% set of real numbers
\newcommand{\Rbb}{\ensuremath{\mathbb R}}

\usepackage{palatino}
\usepackage[sc]{mathpazo}

\usepackage{etoolbox}
\AtEndEnvironment{subequations}{\setcounter{mysub}{0}}

\newcounter{mysub}
\setcounter{mysub}{0}
\renewcommand{\themysub}{def\arabic{mysub}}

\begin{document}
\noindent
For any real symmetric matrcies $\Abf$ such that $\rank(\Abf_{n\times n})=r$,
the following statements are equivalent
and any one of them can serve as the definition of
\emph{positive semi-definite} (\psd) matrices.
\begin{subequations}\label{eq:1}
\begin{itemize}
  \item $\trans\xbf \Abf\xbf \geq 0$ for any non-zero vector
    $\xbf\in\Rbb^{n\times 1}$.
    \hfill\refstepcounter{mysub}\textup{(\themysub)}\label{eq:1a}%
  \item All the $n$ eigenvalues of $\Abf$ are non-negative.
    \hfill\refstepcounter{mysub}\textup{(\themysub)}%
  \item $\Abf=\trans\Bbf \Bbf$ for some $\Bbf$ with $\rank(\Bbf)=r$.
    \hfill\refstepcounter{mysub}\textup{(\themysub)}%
\end{itemize}
\end{subequations}

\noindent
We refer to the global equation \eqref{eq:1} and to subequation \eqref{eq:1a}

\end{document} 

Output:

Answer 2

May be this is what you want but not sure.

\begin{align}
& \hspace*{-3cm} \bullet \;\; \trans\xbf \Abf\xbf \geq 0 \text{  for any non-zero vector } \xbf\in\Rbb^{n\times 1}.\tag{def1} \label{ok} \\
& \hspace*{-3cm} \bullet \;\;  \text{ All the } n\text{ eigenvalues of } \Abf  \text{are non-negative}.\tag{def2} \label{ok2} \\
& \hspace*{-3cm} \bullet \;\;  \Abf=\trans\Bbf \Bbf  \text{ for some } \Bbf \text{ with } \rank(\Bbf)=r. \tag{def3} \label{ok3} 
\end{align}
This is Eq.\eqref{ok} and not quite Eq.\eqref{ok2}