# Any connection between the trace operator and the trace of a matrix?

We have the concept of trace operator in functional analysis and PDE theory at wiki. My first question is, as the informal discussion stated, if $$u$$ is a function that is $$C^1$$ (or even just continuous) on the closure $$\bar \Omega$$ of $$\Omega$$, i.e., $$u \in C^1(\bar\Omega)$$, its function restriction, say $$u|_{\partial \Omega}$$, is well-defined and continuous on $$\partial \Omega$$. Then it noted: "Remember that $$\Omega$$ is an open set, so boundary points are not in $$\Omega$$ and consequently $$u$$ itself is not defined at boundary points." I don't understand why $$u$$ is undefined at $$\partial \Omega$$ if $$u \in C^1(\bar\Omega)$$.

My second question is about the idea of trace operator linked to the trace of matrix. I first learned the term of trace from linear algebra that is defined for the sum of diagonal elements of a matrix. Is there any connection between the trace of a matrix and the trace operator of a function in Sobolev space?

Before that sentence, the article discusses functions $$u:\Omega\to \mathbb R$$, and says the trace is the limit $$\lim_{x\to x_0, \ x\in \Omega} u(x)$$ with $$u_0\in \partial \Omega$$ (if it exists). If this limit exists then everything is fine (i.e., $$u$$ continuous from the closure of $$\Omega$$ to $$\mathbb R$$, or $$u$$ uniformly continuous from $$\Omega$$ to $$\mathbb R$$).