I found a lot of information on the Internet but still can't fully understand the geometric meaning of each step of PCA.
I probably know the application of PCA but I don't know enough about the meaning of this step, but I hope to fully understand the logic and meaning of each step of PCA from beginning to end.
Suppose the centered matrix is a 2*2 matrix X
Step
1.Get the centered matrix=>move the coordinate origin (0.0) to the center of the data group
2. Multiply the transposed matrix of the centered matrix with the centered matrix =>?
X^T*X
I know that the meaning of matrix multiplication is the area between two vectors, but I don't quite understand the meaning of this step. I know diagonal are variance.
3. Take eigenvalues&eigenvectors for the product of centered matrix & its transposed matrix =>?
I only know eigenvalue represent scale & direction of eigenvectors and eigenvector are (x,y).
4. Arrange the unit eigenvectors from left to right according to the value of eigenvalues to get a new matrix W
5.PCA=XW=>Convert the data of X to a new span?Maybe?
6. PCA^T * PCA=>?
I remember that PCA seems to have a sentence of maximizing variance, but I can't understand how it is applied to it.
I need an explanation of the geometric meaning of step 2, 3, 4, 5, 6.
Thank you very much!
Last edited: Feb 21, 2023