\[...\] is the preferred way to initiate/set a display math equation, rather than $$...$$. See Why is \[ ... \] preferable to $$ ... $$?.
Two things you need to change in order for your definition to compile:
You need to set math content in math mode, so use $...$ (or \(...\)).
Vector notation uses [...], not \[...\], which is used for a display math equation (see above).
You're probably after
and $\pmb{\pi}(t) = [ \pi_0(t), \pi_1(t), \dots, \pi_{n-1}(t) ]^T$.
As an aside, keep the document content and structure together by not leaving a blank line between the preceding and following text for a display equation:
Let $n \times n$ \emph{Hessenberg-Matrix} $\pmb{H}_n$ with coefficients $\beta_j^k$ be
\[
\pmb{H_n} = \begin{pmatrix}
\beta_0^0 & \beta_1^1 & \cdots & \beta_{n-1}^{n-1} \\
1 & \ddots & \quad & \vdots \\
\vdots & \ddots \\
0 & \dots & 1 & \beta_0^{n-1}
\end{pmatrix}
\]
and $\pmb{\pi}(t) = [ \pi_0(t), \pi_1(t), \dots, \pi_{n-1}(t) ]^T$.
\pmb,\bmfrom thebmpackage is recommended. – egreg Jul 09 '18 at 21:14