Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The rank is the number of linearly independent rows/cols of a matrix. Generally, we think of linear independence as a binary property. But we could imagine an alternative definition that allows for numbers in the range [0,1]. Then, we could have fractional ranks.

I'm curious if there's any use to such generalizations of rank/independence, or if anyone has even thought about it before?

share|improve this question
2  
Dimensions of objects in monoidal categories can be arbitrary elements of the base ring, and sometimes this freedom is used: mathoverflow.net/questions/16668/… –  darij grinberg Jan 31 at 1:32
    
Hello Mike -- You ask first for every number in [0,1] but then for fractional ranks. Wikipedia let's a fractional part be real, but the word made me think of fractions. Are you interested in situations in linear algebra where the generalized ranks are rational? –  Terry Loring Jan 31 at 1:50
    
Sorry, the number between [0,1] would be an alternative definition for two rows being linearly independent. Then the rank could take on any real range from [1:n], where n is the number of rows/col of the matrix. That's just one idea I had about how ranks might be generalized. You could potentially have others, e.g. by allowing fractionally many columns in the matrix, but I this seems to make even less sense intuitively. –  Mike Izbicki Jan 31 at 2:26
add comment

1 Answer

up vote 8 down vote accepted

von Neumann thought about this; the keyword is continuous geometry.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.