I have an annoyance reflex when people talk about things like "the derivative of a matrix". A matrix is a notational concept, not an object in itself. It makes as much sense to me as saying the derivative of a set, or the derivative of an array (i.e. as opposed to a vector).
It should be derivatives "with" matrices, not "of", in my mind.
Not that it matters in practice, but ... if there's one field where precision of language matters, it should have been mathematics. So it bothers me.
A derivative is a derivative of a function (or a function like object, say a functional). The phrase "the derivative of a matrix" keeps it unclear whether "the matrix" is the function, or whether "the matrix" is a variable, an element in the domain of the function.
Bereft of that information, the phrase is as meaningful as "the derivative of 4.2"
The complete statement is: derivative of a matrix expression or derivative of an expression involving matrices. But this is usually clear from the context.
Yes; like I said, it's a reflex. I know that it's abuse of terminology and I shouldn't overthink it, but, there's always a bit of a nag in my head when I hear it.
I have a similar annoyance with the notation for probability. I think stuff like `p(a)` is a historical mistake. We would have been better served with a notation that would make clear the distinction between a domain as an entity in itself, the set of possible values in the domain, and the action/event of selecting one or more values from that domain.
But oh well. The naming problem is one of the hardest problems in science after all :)
No, it's a collection of vectors (which may be considered ordered or unordered depending on context). The 'vectors' part is fine, but the 'collection' part is the bit that causes the confusion. You can't differentiate a collection (even of vectors), any more you can differentiate a bag of coloured balls. What's the derivative of 'blue'?
But if that collection is plugged into an expression, so as to denote a collection of expressions when result when the individual element vectors are plugged into the expression, and the resulting expressions are differentiable, and you then differentiate each of those expressions and then collect the result back in the same order and represent it as a matrix, then that's fine.
This is what happens when we talk about "matrix differentiation", we mean "differentiation with matrices". But the matrix itself is not a derivable object. Only the expressions upon which it confers its notational semantics have a derivative.
No. It is a mathematical fact that matrices fulfill the conditions of forming a vector space (subject to reasonable conditions like having elements drawn from a field). Matrices are just as differentiable as (other) vectors!
I think you're arguing that a matrix with constant real entries isn't differentiable with the standard derivative. This is true of course, but hasn't anything to do with vectors.
It should be derivatives "with" matrices, not "of", in my mind.
Not that it matters in practice, but ... if there's one field where precision of language matters, it should have been mathematics. So it bothers me.