Is "matrix clause" synonymous with "main clause"? What exactly is a matrix clause?

Question

A lot of people seem to understand "matrix clause" as a synonym for "main clause". For instance, a comment I just chanced upon on a language SE site states:

It's a synonym for main clause (or independent clause) as opposed to subordinate clause (or dependent clause, or indeed embedded clause).

Even some Wikipedia articles seem to echo this:

Main clauses (matrix clauses, independent clauses) are those that can stand alone as a sentence.

That isn't exactly the way I understand the concept. I am under the impression that a matrix clause is a clause with an embedded clause, so it doesn't have to be a main clause, and on the other hand a main clause is not necessarily a matrix clause. Too bad matrix clause doesn't have its own Wikipedia page which would've settled this. But even the name matrix clause is an obvious clue as to its properties. As this discussion demonstrates, matrix comes from "mother" or "a female animal that gives birth to others", pointing to the fact that it is a clause with child clauses. Online sources¹ also confirm my understanding:

A matrix clause is a clause that contains another clause. Thus, the main clause in (37), the professor told the students, is a matrix clause since it contains another clause... So the situation described in the embedded clause is contained by, and functions as an element of, the situation described by the matrix clause.

A matrix clause is often a main clause . . ., but it need not be: it can itself be a subordinate clause.

But it bothers me that my understanding and this definition run counter to that Wiki article. What exactly is a "matrix clause"?

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¹Books cited in this source:

• Martin J. Endley, Linguistic Perspectives on English Grammar. Information Age, 2010
• R.L. Trask, Dictionary of English Grammar. Penguin, 2000

Answer 1

The Cambridge Grammar of the English Language (Huddleston & Pullum 2002) states:

The clause in which a subordinate clause is embedded is called the matrix clause -– in [1vb], for example, subordinate that Liz was ill is embedded within the matrix clause He said that Liz was ill. Subordination is recursive, i.e. repeatable, so that one matrix clause may be embedded within a larger one, as in I think he said that Liz was ill. (p. 47. Bold and italics as in the original)

So (1) below is a main clause:

1. Liz was ill.

It isn't a matrix clause, however, because it contains no subordinate clause.

In (2) below, the whole sentence is a main clause and also a matrix clause and, as described above, that Liz was ill is a subordinate clause.

2. He said that Liz was ill.

In (3) below, the whole sentence is a main cause and a matrix clause:

3. I think he said that Liz was ill.

The sentence contains a subordinate clause, he said that Liz was ill. This subordinate clause has another subordinate clause embedded within it, that Liz was ill. The clause he said that Liz was ill is a matrix clause in relation to the smaller clause that Liz was ill.

We can see therefore that the term matrix clause is in some sense deictic. A clause is a matrix clause specifically in relation to the clause it contains. We could think of it as a 'container'.