is there a Relationship between duplicity of EigenValue and dimension of it's EigenSpace? giving characteristic polynomial of a matrix (Which has eigenvalues with it's duplicity) how can we understand the dimension of eigenspace of each eigenvalue without direct calculation?
in addition, is there a relationship between the power of eigenvalue in minimal polynomial and dim of corresponding eigenSpace?
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at least, Can we discuss about Zero EigenValues and Null Space?

The power of the eigenvalue in the characteristic polynomial is called algebraic multiplicity and the dimension of its eigenspace geometric multiplicity. One can show that 
$$1 \leq \text{ geometric mult.} \leq \text{algebraic mult.}$$
always holds. Note that equality does not hold in general, e.g. take
$$\pmatrix{0 & 0 \\ 1 & 0}$$
which has char. poly. $\chi(\lambda) = \lambda^2$ but the eigenspace corresponding to $\lambda = 0$ is $span \pmatrix{0\\1}$.
For any $n \times n$-matrix $A$ of rank $r < n$, $\lambda = 0$ is an eigenvalue and we have
$$\text{geometric mult.} = \dim \ker A = n-r.$$
However, the algebraic mult. may still be strictly greater than the geometr. mult. (see example above).
