How do I determine the degree of freedom

Question

I have been reading through a paper, in the paper author described a parameter vector $\Theta^{(k)}=(\lambda_1, \cdots, \lambda_k, \sigma^2 , V_1, \cdots, V_k)$ for $k \in \left\{0,1,\cdots, p-1\right\}$. Then author mentions that eigenvalue of complex covariance matrix are real and eigenvectors are complex, hence $\Theta^{(k)}$ has parameters $k+1+2pk$. My question is how do I know that it has above mentioned free paramters? How do I determined the degree of freedom for the above parameter vector?