Sample principal component

Let the columns of a centered design matrix $X\in {\mathbb{R}}^{n\times p}$ be variables and write $X^{T}X=V\Lambda V^T$ by the spectral theorem for real symmetric matrices. The columns $u_i$ of $U=XV$ are the sample principal-component score vectors. They satisfy $u_i^T{u}_j={\Lambda }_{ij}$, so distinct score vectors have zero sample covariance and $u_i$ has sample variance ${\Lambda }_{ii}/n$ under the divisor-$n$ convention.

When ${\Lambda }_{ii}>0$, the normalized sample principal component $w_i=u_i/\sqrt{{\Lambda }_{ii}}$ has Euclidean norm one. The vectors $w_i$ form an orthonormal set spanning the column space of a full-column-rank centered design matrix.