# Orthogonal principal component of submatrix

Consider a data matrix $$M_{n ~ x ~Features}$$ and its top two principal components ($$PC_1$$ and $$PC_2$$). For any of submatrix $$S_{n ~x ~F}$$, where $$F \subset Features$$ (i.e. S contains all samples but a subset of Features), how to find principal components of S that are also orthogonal to $$PC_1$$ and $$PC_2$$?

A heuristic approach might be to find principal components of $$S - W_1 * PC_1 - W_2 *PC_2$$, where $$W_1$$ and $$W_2$$ are linear regression coefficient. Is this correct or any other better/principled approach?

• Please explain what sense of "independence" you are using: is this the probabilistic meaning, where the matrix is random, or perhaps do you mean "linearly independent"? And in that case, please explain how a PC of a submatrix can reasonably be compared to a PC of the main matrix. (This will depend on what you mean by "submatrix.") – whuber Jul 24 '19 at 13:11
• @whuber Yes, I meant linearly independent (i.e. orthogonal). Also for the sake of simplicity lets assume M is normally distributed. I also clarified the submatrix S contains all samples but only a subset of features. – avi Jul 24 '19 at 17:41
• Thank you. The answer to your modified question is "yes." – whuber Jul 24 '19 at 20:43