There needs to be a blank line before the table header for it to render correctly in the final output. Unfortunately, the table renders properly in the preview even without a blank line before the header, so this often causes confusion. This issue was reported on Meta SE here: Markdown table preview does not detect missing blank line before table header.
Here is revision 2 of the post under discussion; it does not have a blank line before the header. Everything renders fine in the preview, but the final output is broken.
After being introduced to the Jordan normal form I wanted to see, if you could classify all 3x3 Matrices by their Jordan normal form and how they would look. This is the table I arraived at, however I was a bit confused, because all Matrices with same minimal polynomial and characteristic polynomial are also similar in this classification. Does that hold for 3x3 Matrices of $\mathbb{R}$? If not, where did I go wrong?
 $\chi_A$  $\mu_A$  partition  jordan 
::::
 $(Xa)^3$  $(Xa)$  $(1,1,1)$  $Diag(a,a,a)$ 
  $(Xa)^2$  $(2,1)$  $Diag(J_2(Xa),a)$ 
  $(Xa)^3$  $(3)$  $J_3(Xa)$ 
 $(Xa)^2\cdot (Xb)$  $(Xa)\cdot (Xb) $  $(1,1)(1)$  $Diag(a,a,b)$ 
  $(Xa)^2\cdot (Xb)$  $(2)(1)$  $Diag(J_2(Xa),J_1(Xb)$ 
 $(Xa)\cdot (Xb) \cdot (Xc)$  $(Xa)\cdot (Xb)\cdot (Xc)$  $(1)(1)(1)$  $Diag(a,b,c)$ 
 $(Xa)\cdot (X^2+bX+c)$  $(Xa)\cdot (X^2+bX+c)$  $(1),(1)$  $Diag(J_1(Xa),J_1(X^2+bX+c))$ 
Here is the same content but with a blank line added before the header:
After being introduced to the Jordan normal form I wanted to see, if you could classify all 3x3 Matrices by their Jordan normal form and how they would look. This is the table I arraived at, however I was a bit confused, because all Matrices with same minimal polynomial and characteristic polynomial are also similar in this classification. Does that hold for 3x3 Matrices of $\mathbb{R}$? If not, where did I go wrong?
$\chi_A$ 
$\mu_A$ 
partition 
jordan 
$(Xa)^3$ 
$(Xa)$ 
$(1,1,1)$ 
$Diag(a,a,a)$ 

$(Xa)^2$ 
$(2,1)$ 
$Diag(J_2(Xa),a)$ 

$(Xa)^3$ 
$(3)$ 
$J_3(Xa)$ 
$(Xa)^2\cdot (Xb)$ 
$(Xa)\cdot (Xb) $ 
$(1,1)(1)$ 
$Diag(a,a,b)$ 

$(Xa)^2\cdot (Xb)$ 
$(2)(1)$ 
$Diag(J_2(Xa),J_1(Xb)$ 
$(Xa)\cdot (Xb) \cdot (Xc)$ 
$(Xa)\cdot (Xb)\cdot (Xc)$ 
$(1)(1)(1)$ 
$Diag(a,b,c)$ 
$(Xa)\cdot (X^2+bX+c)$ 
$(Xa)\cdot (X^2+bX+c)$ 
$(1),(1)$ 
$Diag(J_1(Xa),J_1(X^2+bX+c))$ 
The above post uses colons :
for aligning the content of the columns, but the final output looks terrible. Unfortunately, MathJax has been broken like this for quite some time, ever since the switch to the responsive design. I listed some of these bug reports in a comment elsewhere. Honestly, I despair of this ever getting fixed.
User watertrainer's choice to use "displaystyle" instead of the colons is a decent workaround in this case. But, the Markdown rendering of the table was fixed just by adding a blank line before the table header, and that had nothing to do with the "inline" math mode in the table cells.