Add .is_singular method to matrices

[rbeezer]

Some approaches to teaching linear algebra define nonsingular matrices before broaching matrix invertibility. This will just add is_singular, with some changes to the associated documentation. thus making it a bit easier for Sage usage to follow one possible development of the topic.

[jason]

Do you think it might be better to add the positive phrasing, m.is_singular(), rather than the negative nonsingular version?

[rbeezer]

Replying to jason:

Do you think it might be better to add the positive phrasing, m.is_singular(), rather than the negative nonsingular version?

I had that thought. It would then no longer be an alias, which means (a) the documentation could be separate from invertibility, and (b) it would only need to be done once (is_invertible is defined twice). Does it work for users/students? I like to define nonsingular, but there is no real good reason for that, I guess. What's your preference?

[jason]

My advisor (who was a very good writer) taught me that when you are talking about matrices with determinant zero, and that is the focus, you should use "singular", and when you are talking about matrices with nonzero determinant, and that is the focus, you should use "invertible". This was to avoid confusing the reader with too many negatives. So I think having m.is_singular() and m.is_invertible() is a nice combination. It's easy to say "to check if a matrix is nonsingular, see if m.is_singular() is false".

Personally, I use "invertible" when talking about invertible matrices, so I guess it isn't that huge of a deal. I just thought things could be clearer by not putting a negatively-phrased is_* method in there.

[rbeezer]

Replying to jason:

Sounds very good to me. I'll likely take this approach.

[jason]

I guess the other philosophy at play here is that typically, is_* functions determine what an object *is*, not what it is not. There probably are exceptions, though...

[rbeezer]

It is never what it seems. Over a commutative ring, the right thing to do is to define is_singular() to be the condition that the matrix has a zero determinant.

So this needs to be a new method, it is better stated as the positive version, and it is more than just a convenience.

[robertwb]

Looks good to me.