# Do the principal components change if we apply PCA more than once (recursively) on data?

Consider a set $X=(X_1; \dots; X_n)$ of $n$ data points such that $X_i \in \mathbb{R}^d$ is a column vector. Let $Y = \text{pca_proj}(X)$ denote the projection of points in $X$ according to the PCA components i.e. $$Y_i \in \mathbb{R}^d\\ Y_i = W X_i$$ where $W \in \mathbb{R}^{d \times d}$ is the projection matrix obtained from PCA, and each row of $W$ is an eigen vector of $X^T X$.

Now let's define $Z = \text{pca_proj}(Y)$. My questions is: does $Y = Z$? If not, what happens if we keep applying PCA recursively on a matrix? Does it converge to a specific matrix?

• Since PCA can be (and usually is) defined without reference to any particular basis of $\mathbb{R}^d$, the answer is immediate. If that's not perfectly clear, then you might enjoy reading over many of the answers at stats.stackexchange.com/questions/2691 .
– whuber
Mar 10 '16 at 22:22
• Subjecting PCs of PCA again to PCA isn't helpful because PCs are orthogonal (uncorrelated) variables: the covariance matrix is diagonal. Mar 10 '16 at 22:30