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We consider position-dependent mass (PDM) Lagrangians/Hamiltonians in theirstandard textbook form, where the long-standing \emph{gain-loss balance}between the kinetic and potential energies is kept intact to allow conservationof total energy (i.e., $L=T-V$, $H=T+V$, and $dH/dt=dE/dt=0$). Under suchstandard settings, we discuss and report on $n$-dimensional PDM damped harmonicoscillators (DHO). We use some $n$-dimensional point canonical transformationto facilitate the linearizability of their $n$-PDM dynamical equations intosome $n$-linear DHOs' dynamical equations for constant mass setting.Consequently, the well know exact solutions for the linear DHOs are mapped,with ease, onto the exact solutions for PDM DHOs. A set of one-dimensional anda set of $n$-dimensional PDM-DHO illustrative examples are reported along withtheir phase-space trajectories.

$n$-dimensional PDM-damped harmonic oscillators: Linearizability, and exact solvability