The original matrix $X$ (dimension $n \times p$, $n$ observations, $p$ variables) is transformed to $Y$ by centering and/or standardizing the columns. Using the singular value decomposition (SVD), we can write $$Y = UDV^T=\sum_{k=1,...p}d_k\mathbf u_k\mathbf v_k^T,$$ where the $\mathbf u_k$ are $n$-dimensional column vectors, the $\mathbf v_k$ are $p$-dimensional column vectors, and the $d_k$ are a non-increasing sequence of non-negative scalars. The biplot is formed from two scatterplots that share a common set of axes and have a between-set scalar product interpretation. The first scatterplot is formed from the points ($d_1^\alpha u_{1i}, d_2^\alpha u_{2i}$), for $i = 1,...,n$. The second plot is formed from the points ($d_1^{1-\alpha}v_{1j}, d_2^{1-\alpha}v_{2j}$), for $j = 1,...,p$. The $\alpha$ can be set as 0, 0.5, or 1. [Wikipedia]