Why do some matrices do PCA to get a few rows of matrices?

Many times there will be a n*m matrix as a PCA (M-dimensionality) and then get a m* (M-1) matrix such a result. Before it was mathematically deduced to get this conclusion, however,

See a very vivid explanation today:

Consider what PCA does. Put simply, PCA (as most typically run) creates a new coordinate system by (1) shifting the origin to the centroid of your Data, (2) squeezes and/or stretches the axes to make them equal in length, and (3) rotates your axes into a new Orientati On. (For more details, see this excellent CV thread:making sense of principal component analysis, eigenvectors & Eigenval UEs.) However, it doesn ' t just rotate your axes any old. Your New X1 (the first principal component) is oriented in your data ' s direction of maximal variation. The second principal component is oriented in the direction of the next greatest amount of variation it is Orthogona Lto the first principal component. The remaining principal components is formed likewise.

With the examine @amoeba ' s example. Here are a data matrix with points in a three dimensional space:

X=[1 2 1 2 1 2 "

Let's view these points in a (pseudo) three dimensional scatterplot:

So let ' s follow the steps listed above. (1) The origin of the new coordinate system would be located at(1.5,1.5,1.5) . (2) The axes is already equal. (3) The first principal component would go diagonally from(0,0,0) To(3,3,3) , which is the direction of greatest variation for these data. Now, the second principal component must is orthogonal to the first, and should go in the direction of the greatestremainingVariation. But what direction was that? Is it from(0,0,3) To(3,3,0) , or from(0,3,0) To (3,0< Span id= "mathjax-span-166" class= "Mo" >,3 ) , or something else?  there is no remaining variation, so there cannot being any further principal components .

with n=2 data, we can fit (in most) N−1=1 principal components.

Why do some matrices do PCA to get a few rows of matrices?