The matrix rank is computed based on the number of singular values that are larger than a particular floating point tolerance. This means that when a singular value happens to be very near that cutoff, the rank will be sensitive to floating point roundoff errors. In general when working with floating point math, computing the "same" value in different ways will often lead to results that are not equivalent. A classic example is this:

>>> 0.1 + 0.2 == 0.3
False

I suspect in your case, you have singular values which are right at the edge of the cutoff, meaning that when you compute the "same" result two different ways, different floating point roundoff leads to values on either side of the cutoff. You can adjust for this by passing a larger tol parameter to np.matrix_rank; see the np.matrix_rank docs for details on its default value.

Thanks for your answer, I' ve already read your book (it is one of my favourite!). Passing the same tolerance helped, but I cannot understand why when the default value is used (which is the same , because types of arrays are the same on both sides), the result is different. Reformulating question why when you pass the exact tol value result is correct, and when using default - not.
Your "classic" example is teachful, but how is it connected with my problem, the lowest singular value is rather big - larger than 0.01, could there be problems with roundoff ?

Reformulating question why when you pass the exact tol value result is correct, and when using default - not.

Neither is correct and neither is incorrect: what matrix_rank does is look at the singular values and count the number above some cutoff. You're computing two slightly different sets of singular values, so you get a different answer. If what you're computing is sensitive to that difference, you need a looser tolerance to reflect that.

Your "classic" example is teachful, but how is it connected with my problem, the lowest singular value is rather big - larger than 0.01, could there be problems with roundoff ?

Sorry to be unclear. My point was to illustrate that if you compute the "same" value two different ways in floating point arithmetic, you may not get the exact same answer. You are (implicitly) computing the "same" singular values two different ways, and getting different answers. This is expected in floating point arithmetic, because all computations are subject to floating point roundoff error, and different ways of expressing the same computation will lead to errors accumulating differently.

Does that make sense?

Thank you very much, I appreciate your help