How do we know that this matrix product produces a symmetric matrix? I'm following a derivation in a textbook and I'm getting tripped up on one of the steps. $O$ is an objective function that we're trying to minimize, $V$ is a $k\times1$ vector, $k$ is a positive integer, $U$ is a $k\times2$ matrix, $p$ is a $2\times1$ vector, and $^T$ is the matrix transpose operation. In the text, they go from $$ O = (V - Up)^T(V - Up) \,,$$ to $$ O = V^TV - 2V^TUp + p^TU^TUp \,.$$ The part I'm stuck on is the middle term: $2V^TUp$. When I expand the original expression I get $$ O = V^TV - p^TU^TV - V^TUp + p^TU^TUp\,.$$ The only way that I can see them arriving at this result is if $V^TUp$ is symmetric, i.e. $$V^TUp = (V^TUp)^T = p^TU^TV \,.$$ However, the author doesn't provide any statement or proof that this is the case. My background is in engineering not math, so maybe it's obvious to a mathematician. **How do we know that $V^TUp$ is symmetric?**

$V^TUp$ is symmetric because it's a $1 \times 1$ matrix!