为什么Sprank（A）和A \\ b在Matlab中报告不同的排名？

Question

I have a point set P and I construct it's adjacent matrix A by k-nearest neighbor. Each row of A is [...+1...-1...], indicates a pair of neighbor points. The size of A is 48348 x 8058, sprank(A) is 8058. But when I do the following, it gives me a warning: "Warning: Rank deficient, rank = 8055, tol = 8.307912e-10."

a=A*b; c=A\\a;

and norm(cb) is quite large. It seems something is wrong with the adjacent matrix A, but I can't figure it out. Thanks in advance!

Answer 1

sprank only tells you how many rows/columns of your matrix have non-zero elements, while A\\b is reporting the actual rank of the matrix which indicates how many rows of your matrix are linearly independent . For example, for following matrix:

A = [-1  1  0  0;
      0  1 -1  0;
      1  0 -1  0; 
      0  0  1 -1]

sprank(A) is 4 but rank(A) is only 3 because you can write the third row as a linear combination of the other rows, specifically A(2,:) - A(1,:) .

The issue that you need to address is either in how you're computing A (if you expect that to generate a system of linearly independent equations) or you need to find a way to use A that doesn't require factorizing a rank deficient matrix.