For $k=1,\dots,K$ with $K\ge 2$, let $\boldsymbol{v}_k$ be a $r_k$-dimensional random vector whose mean is zero and covariance matrix is the identity matrix ${\bf I}_{r_k\times r_k}$. Also $span(\boldsymbol{v}_k^\top)\ne span(\boldsymbol{v}_{k'}^\top)$ for any $k\ne k'$. Here $span(\boldsymbol{v}_k^\top)$ means the space spanned by entries of $\boldsymbol{v}_k$.
$\boldsymbol{v}_1,\dots,\boldsymbol{v}_K$ can follow different distributions with unequal $r_k$, and can be correlated with each other.
Must the correlation/covariance matrix of the concatenated vector $(\boldsymbol{v}_1^\top,\dots,\boldsymbol{v}_K^\top)^\top$ have a "nonzero" eigenvalue $\le1$.
Note that in this case, the correlation and covariance matrix of $(\boldsymbol{v}_1^\top,\dots,\boldsymbol{v}_K^\top)^\top$ are the same, and can be low-rank.
I know this is true for $K=2$, but what if $K>2$?
Not necessarily. And even if $K=2$, the correlation matrix might be $\begin{pmatrix}1&1\\1&1\end{pmatrix}$, which has eigenvalues 2 and 0.