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 Homework Statement

Full disclaimer: This post is the same as the one in the following link with some slight modifications (I'm the author of the original) https://physics.stackexchange.com/questions/469386/understandingsolutionsofthediracequation
In one of the lectures that I'm currently taking we encountered the Dirac equation. The general solution was given as
$$\psi ( x ) = \sum _ { s } \int \frac { d ^ { 3 } \bf { p } } { ( 2 \pi ) ^ { 2 } 2 \omega _ { p } } \left[ a _ { s } ( p ) u ^ { s } ( p ) e ^ {  i p \cdot x } + b _ { s } ^ { * } ( p ) v ^ { s } ( p ) e ^ { + i p \cdot x } \right],$$
where
$$u^{s}(p)=\begin{pmatrix}{\sqrt{\sigma \cdot p} \xi^{s}} \\ {\sqrt{\overline{\sigma} \cdot p} \xi^{s}}\end{pmatrix} \quad\text{and}\quad v ^ { s } ( p ) = \begin{pmatrix} { \sqrt { \sigma \cdot p } \xi ^ { s } } \\ {  \sqrt { \bar { \sigma } \cdot p } \xi ^ { s } } \end{pmatrix}.$$
Note that we defined ##\sigma^\mu \equiv (1,\vec{\sigma})## and ##\bar\sigma^\mu \equiv (1,\vec\sigma)## and ##s\in\{+,\}## for
$$\xi^+ \equiv \begin{pmatrix}1\\0\end{pmatrix},~\xi^\equiv\begin{pmatrix}0\\1\end{pmatrix}.$$
My problem is now that I'm a bit confused on how to evalute the expression ##\sqrt{p\cdot\sigma}\xi^s##. If I understood correctly we have ##p\cdot \sigma = p_\mu\sigma^\mu## which makes this expression a matrix. But how am I supposed to take the squareroot now?
This expression can also be found in Perskin chapter 3, S. 46.
 Homework Equations
 All given above.
some notes:
There was actually no proof given why ##u^s(p)## or ##v^s(p)## should solve the Dirac equation, only a statement that one could prove it using the identity
$$(\sigma\cdot p)(\bar\sigma\cdot p)=p^2=m^2.$$
We were using the Welyrepresentation of the ##\gamma##matrices, if this should be relevant. I think I can prove the statement if someone could explain to me how one should evaluate the expression ##\sqrt{p_\mu\sigma^\mu}\,\xi^s##. In Perskin QFT book I found the explanation
"where it is understood that in taking the square root of a matrix, we take the positive root of each eigenvalue."[pp. 46]
but honestly I can't figure out how this is supposed to work for ##p_1\neq p_2\neq 0##.
I'd like to stress here that I'm not intrested in alternative ways of expressing solutions, I'd like to understand why we can write them in this specific form..
There was actually no proof given why ##u^s(p)## or ##v^s(p)## should solve the Dirac equation, only a statement that one could prove it using the identity
$$(\sigma\cdot p)(\bar\sigma\cdot p)=p^2=m^2.$$
We were using the Welyrepresentation of the ##\gamma##matrices, if this should be relevant. I think I can prove the statement if someone could explain to me how one should evaluate the expression ##\sqrt{p_\mu\sigma^\mu}\,\xi^s##. In Perskin QFT book I found the explanation
"where it is understood that in taking the square root of a matrix, we take the positive root of each eigenvalue."[pp. 46]
but honestly I can't figure out how this is supposed to work for ##p_1\neq p_2\neq 0##.
I'd like to stress here that I'm not intrested in alternative ways of expressing solutions, I'd like to understand why we can write them in this specific form..