documentclass{article}
usepackage{stackengine}
stackMath
newlengthmatfield
newlengthtmplength
defmatscale{1.}
newcommanddimbox[3]{%
  setlengthmatfield{matscalebaselineskip}%
  setbox0=hbox{vphantom{X}smash{#3}}%
  setlength{tmplength}{#1matfield-ht0-dp0}%
  fboxrule=1ptfboxsep=-fboxrulerelax%
  fbox{makebox[#2matfield]{addstackgap[.5tmplength]{box0}}}%
}
newcommandraiserows[2]{%
   setlengthmatfield{matscalebaselineskip}%
   raisebox{#1matfield}{#2}%
}
newcommandmatbox[5]{
  stackunder{dimbox{#1}{#2}{$mathbf{#5}$}}{scriptstyle(#3times #4)}%
}
parskip 1em
begin{document}
$renewcommandmatscale{.6}
matbox{7}{4}{I}{J}{X} =
matbox{7}{2}{I}{R}{T} raiserows{2.5}{matbox{2}{4}{R}{J}{P^T}} +
matbox{7}{4}{I}{J}{E}$
end{document}

$renewcommandmatscale{.05}
matbox{300}{75}{I}{J}{X} =
matbox{300}{25}{I}{R}{T} raiserows{137.5}{matbox{25}{75}{R}{J}{P^T}} +
matbox{300}{75}{I}{J}{E}$

documentclass[12pt]{article}
usepackage[margin=1cm,paper size={20cm,5cm}]{geometry}
usepackage{amsmath,amssymb}

thispagestyle{empty}
begin{document}
[
{begin{pmatrix}
  &    & \
  & X &\
  &     &
end{pmatrix}
mkern-10mu}_{I times J}=
{begin{pmatrix}
phantom{T}\
makebox[20pt][c]{$T$}\
phantom{T}
end{pmatrix}
mkern-10mu}_{I times R}
{begin{pmatrix}
  &    & \
  & raisebox{10pt}{$P^T$} &\
end{pmatrix}mkern-10mu}_{R times J}+
{begin{pmatrix}
  &    & \
  & E &\
  &    &
end{pmatrix}
mkern-10mu}_{I times J}
]

end{document}

documentclass{article}
usepackage{amsbsy}
newcommand*{clap}[1]{hbox to 0pt{hss#1hss}}
newcommand*{mat}[1]{boldsymbol{mathrm{#1}}}
newcommand*{subdims}[3]{clap{raisebox{#1}[0pt][0pt]{$scriptstyle(#2 times #3)$}}}
fboxrule=1pt

begin{document}
Huge
[
framebox[2.5cm]{clap{raisebox{0pt}[1.5cm][1.5cm]{$mat X$}}subdims{-2.5cm} I J} =
framebox[1.5cm]{clap{raisebox{0pt}[1.5cm][1.5cm]{$mat T$}}subdims{-2.5cm} I R}
framebox[2.5cm]{clap{raisebox{5mm}[1.5cm]{$mat P^T$}}     subdims{-1cm} R J} +
framebox[2.5cm]{clap{raisebox{0pt}[1.5cm][1.5cm]{$mat E$}}subdims{-2.5cm} I J}
]

end{document}