$\begin{align} I & = & PP^* \\ & = & \begin{pmatrix} \bf{x_1} & \bf{x_2} &\dots & \bf{x_n} \end{pmatrix} \begin{pmatrix} \bf{x_1^*} \\ \bf{x_2^*} \\ \vdots \\ \bf{x_n^*} \end{pmatrix},\tag{1} \\ & = & \bf{x_1}\bf{x_1^*} + \bf{x_2}\bf{x_2^*} + \dots + \bf{x_n}\bf{x_n^*} , \tag{2} \end{align}$
where the column vectors of $P$ are an orthornomal basis of $\mathbb{C}^n$ and $P^*$ is the Hermitian conjugate of $P$.
I have no idea how to go from (1) to (2). From (1), all I know is that the product can also be expressed in terms of the complex inner product of the the rows of $P$ and the columns of $P^*$ but this does not seem to help and furthermore, the answer is still a matrix and I don't see how it can be expressed as sums. I know that the columns of $P$ form and orthornomal basis means that the complex inner product of any two columns of $P$ will be 0 but I don't yet see where that could come in handy. Please show me the steps to get to $(2)$.