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In the following paragraph, is it appropriate to use the term "transversely" to describe something that has the opposite effect?
Tests have shown that the lower the range, the more likely that a submatrix will be singular. Transversely, the larger the range, the least likely that a submatrix will be singular.
If not, is there another word that I can use instead? I know that, in this instance, I cannot use the word "conversely" as this means something totally different.
No, transversely is incorrect. You might consider "In contrast ..." or "On the other hand...".
I would use "inversely", as in "inversely proportional".
If we have a statement:
The lower the range, the more likely that a submatrix will be singular.
the logical converse of the statement is:
The more likely that a submatrix will be singular, the lower the range.
and the logical inverse of the statement is:
The larger the range, the less likely that a submatrix will be similar.
So I would say that, if your aim is logical correctness, you should use the adverb inversely. This usage is also compatible with the common English meaning of the word inversely. The adverb transversely is just wrong here. From Merriam-Webster online dictionary:
Transversely: in a line or direction running from corner to corner.
But never use the adverb contrapositively. It doesn't exist.
Conversely is correct. In this context, "conversely" does not have the same meaning it has in logic. It most strongly means a claim that reverses one or more aspects of a previous claim. It has a weaker meaning akin to "on the other hand".
My dictionary has this example, "We provide tech-support 24-hours a day, seven days a week; conversely, our competitors only provide tech-support 8-hours a day, five days a week." I personally wouldn't use "conversely" that way and reserve it for cases where the order or direction of something in a causal chain is reversed. But it is not incorrect.
There is nothing wrong with something like this: "Study more and you will get higher grades; conversely, if you study less your grades will surely suffer." The direction of the amount of studying and the movement of the grades are reversed. So "conversely" works perfectly. So long as there is some sense of opposition, it flows quite naturally.
I would simply use and - the problem with using words like transversely or inversely is that they may have very specific meanings in the topic of matrix maths which could cause confusion.
Even if they don't people might think they have a technical meaning and be trying to interpret it.
I think the most appropriate word is correspondingly.
Substituting X for range is lower, and Y for submatrix is likely to be singular, we have
If X is true then Y is true. Correspondingly, if X is not true then Y is not true.
Although the second statement doesn't logically have to follow from the first, this will often be the case for many non-trivial propositions of the form "If X then Y". A shorter way of asserting both is "If and only if X, then Y".
Although it seems odd, I also endorse @David Schwartz's conversely, despite the fact that superficially this word is the direct opposite of correspondingly.
If we can use an expression rather than a single word, I'd say by the same token, which I think implies a relationship between the two statements without getting bogged down in whether it's inverse, converse, or corresponding.
You don't want "transversely" or "conversely" or anything else. Those two sentences mean exactly the same thing! If you really feel the need to repeat yourself, I suppose you could write "To put it another way..."
