Let G be a finite group. $\hat{G}$ donates a set of all irreducible representations of G over $\mathbb{C}$. $C_G(h)$ denotes centraliser of an element $h\in G$. I want to prove $|C_G(h)|=\sum\limits_{\chi \in \hat{G}}\chi(h)\overline{\chi (h)}$.
I dont know how to get this. Please help me with this.
One can approach this by phrasing the character orthonormality in terms of matrices. So we will view characters as complex valued functions on conjugacy classes of $G$, and note that the orthonormality of irreducible characters can be expressed as: \begin{equation}\sum_{C}|C|\chi_V(C)\overline{\chi_W(C)}=|G|\delta_{V,W}\end{equation}
Where $V$, $W$ are irreducible representations, and $C$ runs over the conjugacy classes of $G$. Let $M$ be the character table, viewed as a $\text{#}C\times \text{#}C$ matrix. Then if $D$ is the diagonal matrix with $(C,C)$ entry equal to $|C|$, this equality becomes \begin{equation} MDM^*=|G|I\end{equation}
Where $M^*$ is the conjugate transpose of $M$. So rearranging this, we obtain \begin{equation}M^*M=|G|D^{-1}\end{equation}
Noting that $\frac{|G|}{|C|}$ is the cardinality of the centraliser of an element of $C$ yields your result by comparing entries of these two matrices.