Short Answer: The principal components are the eigenvectors of the correlation matrix. Therefore, each principle component $(V)$ multiplied by the correlation matrix $(C)$ will give us the same correlation matrix times the corresponding eigenvalue $\lambda$:
CV = \lambda V
Details: Given n-dimensional data ($x_i \in R^n$), suppose we have $m$ datapoints represented as rows in a matrix $X$ (An $ m\times n$ matrix). Given that $Cor(i,j)$ is the correlation of 2 dimensions $i$ and $j$, the correlation matrix is defined as:
Cor(0,0) & Cor(0,1) & \cdots & Cor(0, n-1) \\
Cor(1,0) & Cor(1,1) & \cdots & Cor(1, n-1) \\
\vdots & \vdots & \ddots & Cor(n-1, n-1)
Since the correlation matrix is a square matrix of size ($n \times n$), there are n possible eigenvectors for this matrix, and these vectors are the principle components of this data. each principle component $V$ is of size $n \times 1$, and it's corresponding eigenvalue $\lambda$ is a scalar value.