  This paper focuses on the non-asymptotic concentration of the heteroskedastic
Wishart-type matrices. Suppose $Z$ is a $p_1$-by-$p_2$ random matrix and
$Z_{ij} \sim N(0,\sigma_{ij}^2)$ independently, we prove the expected spectral
norm of Wishart matrix deviations (i.e., $\mathbb{E} \left\|ZZ^\top -
\mathbb{E} ZZ^\top\right\|$) is upper bounded by \begin{equation*}
  \begin{split}
  (1+\epsilon)\left\{2\sigma_C\sigma_R + \sigma_C^2 +
C\sigma_R\sigma_*\sqrt{\log(p_1 \wedge p_2)} + C\sigma_*^2\log(p_1 \wedge
p_2)\right\},
  \end{split} \end{equation*} where $\sigma_C^2 := \max_j
\sum_{i=1}^{p_1}\sigma_{ij}^2$, $\sigma_R^2 := \max_i
\sum_{j=1}^{p_2}\sigma_{ij}^2$ and $\sigma_*^2 := \max_{i,j}\sigma_{ij}^2$. A
minimax lower bound is developed that matches this upper bound. Then, we derive
the concentration inequalities, moments, and tail bounds for the
heteroskedastic Wishart-type matrix under more general distributions, such as
sub-Gaussian and heavy-tailed distributions. Next, we consider the cases where
$Z$ has homoskedastic columns or rows (i.e., $\sigma_{ij} \approx \sigma_i$ or
$\sigma_{ij} \approx \sigma_j$) and derive the rate-optimal Wishart-type
concentration bounds. Finally, we apply the developed tools to identify the
sharp signal-to-noise ratio threshold for consistent clustering in the
heteroskedastic clustering problem.
