underbrace under matrix

Below I use some measurements to correctly align the underbrace width with its contents.

documentclass{article}

usepackage{amsmath,xcolor}

newcommand*{vertbar}{rule[-1ex]{0.5pt}{2.5ex}}
newcommandundermat[2]{% http://tex.stackexchange.com/a/102468/5764
  makebox[0pt][l]{$smash{underbrace{phantom{%
    begin{matrix}#2end{matrix}}}_{text{$#1$}}}$}#2}

begin{document}

Original:
begin{align}
    mathbf{W}_{BA} & 
    begin{pmatrix}
  vertbar &  & vertbar & & vertbar &  & vertbar  
  mathbf{x}^B_1 & dots & mathbf{x}^B_1 & dots & mathbf{x}^B_p & dots & mathbf{x}^B_p  
  undermat{k}{vertbar &  & phantom{--}vertbarphantom{--}} & & undermat{k}{vertbar &  & {phantom{--}vertbarphantom{--}}}
  end{pmatrix}
  = notag \
  & begin{pmatrix}
  vertbar &  & vertbar & & vertbar &  & vertbar  
  mathcal{N}(mathbf{x}^B_1)_1 & dots & mathcal{N}(mathbf{x}^B_1)_k & dots & mathcal{N}(mathbf{x}^B_p)_1 & dots & mathcal{N}(mathbf{x}^B_p)_k  
  undermat{k}{vertbar &  & phantom{---}vertbarphantom{---}} & & undermat{k}{vertbar &  & {phantom{---}vertbarphantom{---}}}
  end{pmatrix}.
  
end{align}

% Update vertbar
renewcommand{vertbar}[2]{{%
  setlength{fboxsep}{.5fboxsep}
  ooalign{%
    hidewidthrule[-1.5htstrutbox]{.4pt}{3.5htstrutbox}hidewidthcr%
    colorbox{white}{$#2strut$}rule[-1.5htstrutbox]{0pt}{3.5htstrutbox}%
  }
}}
% Measure some components of the construction
newlength{xpB}settowidth{xpB}{$mathbf{x}_p^B$}
newlength{NxaB}settowidth{NxaB}{$mathcal{N} bigl( mathbf{x}_p^B bigr)_1$}
newlength{NxpB}settowidth{NxpB}{$mathcal{N} bigl( mathbf{x}_p^B bigr)_k$}

Updated:
begin{align}
  mathbf{W}_{BA} &
  Biggl(
    hspace{0.5xpB}
    underbrace{
      hspace{-0.5xpB}
      vertbar{x1B}{mathbf{x}_1^B} hspace{0.5em} cdots hspace{0.5em} vertbar{x1B}{mathbf{x}_1^B}
      hspace{-0.5xpB}
    }_k
    hspace{0.5xpB}
    quad cdots quad
    hspace{0.5xpB}
    underbrace{
      hspace{-0.5xpB}
      vertbar{xpB}{mathbf{x}_p^B} hspace{0.5em} cdots hspace{0.5em} vertbar{xpB}{mathbf{x}_p^B}
      hspace{-0.5xpB}
    }_k
    hspace{0.5xpB}
  Biggr) = notag 
  & Biggl(
    hspace{0.5NxaB}
    underbrace{
      hspace{-0.5NxaB}
      vertbar{Nx1B}{mathcal{N} bigl( mathbf{x}_1^B bigr)_1} hspace{0.5em} cdots hspace{0.5em} vertbar{Nx1B}{mathcal{N} bigl( mathbf{x}_1^B bigr)_k}
      hspace{-0.5NxaB}
    }_k
    hspace{0.5NxaB}
    quad cdots quad
    hspace{0.5NxpB}
    underbrace{
      hspace{-0.5NxpB}
      vertbar{NxpB}{mathcal{N} bigl( mathbf{x}_p^B bigr)_1} hspace{0.5em} cdots hspace{0.5em} vertbar{NxpB}{mathcal{N} bigl( mathbf{x}_p^B bigr)_k}
      hspace{-0.5NxpB}
    }_k
    hspace{0.5NxpB}
  Biggr)
end{align}

end{document}