= Singular-volatility drift arbitrage
{title2=$\sigma^T v=0,\quad v^T\mu>0$}
Work in discounted units with a unit <bank account> and constant asset-gain dynamics $dS=\mu\,dt+\sigma\,dW$. Here $\mu$ is the vector of excess dollar drifts, not an unadjusted physical return. If a holdings vector $v$ satisfies
$$
\sigma^Tv=0,\qquad v^T\mu>0,
$$
finance its initial stock cost through the bank. The resulting zero-cost <self-financing portfolio> has deterministic wealth $t\,v^T\mu>0$ for $t>0$, giving an <arbitrage>.
For $A=\sigma\sigma^T$, absence of such a direction is equivalent to $\mu\in\operatorname{Range}A$. Indeed, $\ker A=\ker\sigma^T$ and $\operatorname{Range}A=(\ker A)^\perp$. If $\mu$ has a nonzero projection onto that <kernel>, taking $v$ to be that projection gives $v^T\mu=\lVert v\rVert^2>0$.
Scaling this deterministic gain makes any unbounded increasing terminal <utility function> unbounded on a positive horizon. A singular <volatility matrix> therefore requires this drift-compatibility check before a finite investment value is asserted. With a nonzero bank rate in undiscounted units, the financing return must first be subtracted.
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