Suppose we have three 5 explanatory variables $X_1, \dots, X_5$. The first principal component is the linear combination $$Y = \epsilon_1 X_1+ \cdots + \epsilon_5 X_5$$
that maximizes $\text{var}(Y)$. Can we write this as $$Y = \mathbf{\epsilon} \cdot \textbf{X}$$
that is in vector notation?
If we wrote both $\mathbf{\epsilon}$ and $\textbf{X}$ as column vectors, would we write it as $$Y = \mathbf{\epsilon}^{T} \cdot \textbf{X}$$
A caveat: $X$ is an $n\times 5$ matrix, where $5$ is the number of features and $n$ are the observations. Then each eigenvector $\epsilon^i$ $i=1,\dots,5$ of the covariance matrix considered in the OP is a vector of length $n$. In this notation $$Y=\epsilon^T X$$
is a $(5\times n)\times (n\times 5)=5\times 5$ matrix (with PCA you get a dimensional reduction of your problem) obtained by the product of matrices. Denoting by $Y^i$ the $i$-th row of $Y$, then
$$Y^i=\left(\epsilon^i\cdot X_1,\dots,\epsilon^i\cdot X_5\right)$$
where the $i$-th eigenvector $\epsilon^i$ is the $i$-th row of $\epsilon^T$ and $X_j$ denotes the $j$-th column of $X$, i.e. the column of the $j$-th feature.