I think you know that lambda_i(M_j) is the ith eigenvalue of M_j. Order them how you like it won't change the identity.

Doesn't he say it here:

where {M_j} is the {n-1 \times n-1} Hermitian matrix formed by deleting the {j^{th}} row and column from {A}.

Yes, he says what M_j is, but he does not say what lambda_i(M_j) is.

Okay, Donald Knuth might be upset that Terry failed to introduce the notion of a spectrum space in a Matrix Analysis paper to an audience that should know about it already.

If I tell you f(x) = y, do you get upset that I don't define the domain, range, codomain, notion of a function, and so forth? Or if I tell you that https:// prepends a good portion of URLs on the world-wide web, would you subject it to the same scrutiny? I hope note -- to require such would be unnecessarily limiting.

My $0.02.

If you tell me that thing about https://, it's not math, it's just a conversation, so why would I care?

If you tell me that f(5) = 10, and then you tell me later on that now I can compute g(x) = f(x) - 4, for all x, from that knowledge, then I surely would wonder what you are talking about.

Just read your original opening statement. I think what everyone is telling you is that you were being unnecessarily hyperbolic towards an author that writes for a mathematical argument. If it offends you so much as you've indicated that you just "moved on," then so be it.

I still stand by my opening statement. Now, after the discussion here I understand what the theorem statement is saying. That's a lot of work I had to do to understand the theorem statement, and it would have been unnecessary, had the theorem been stated "syntactically" correctly.

Now why wasn't the theorem stated syntactically correct? If we had to state it in a mechanised theorem prover like Isabelle then we would extract the definition of the eigenvalues from the statement into a separate definition, like:

For a hermitian n x n matrix H, let lambda_1(A), ..., lambda_n(A) be the eigenvalues of H.

But of course, here we assume an order of the eigenvalues, and we don't really want that, because it is immaterial to what follows. So we rather have to write it like that:

Definition: For a matrix H, let L(H) be the set of its eigenvalues.

Theorem: If H is a n x n Hermitian matrix, then |L(H)| = n.

But, of course, this is not true, as |L(H)| < n is a possibility. So you would have to account for that somehow as well in your definition / notation.

So it is probably better to go back instead again, and explain the notation as follows:

Notation: For a Hermitian n x n matrix H, let lambda_1(H), ..., lambda_n(H) be its eigenvalues (in an arbitrary order).

Theorem: What Tao said.

The notation could still not be written in Isabelle like that, but at least it has proper scoping now.

And I am pretty sure that every single downvote I get for this discussion here is from somebody who doesn't properly understand the theorem statement.

Point of feedback: this assumption does not hold.

Some people also just downvote because they don't like something, although it is right.

True, but this isn't what you asserted.

Nope, it wasn't, thus my amendment.

Without objection.

Cheers.

He says that eigenvalues of A are the lambda_i(A), so it's pretty obvious that lambda_i(M_j) is the i-th eigen value of M_j.

What might not be obvious is the ordering of the eigenvalues, which must be the same for the formula to make sense.

No, the ordering of the eigenvalues of M doesn't matter, because we take their product (or rather, the product of their difference from a fixed number), and multiplication commutes. It's maybe not obvious, and as such I can understand auggierose's complaint to an extent, though it was maybe unnecessarily harshly phrased.

You're right, my bad.

Well, you see, the ordering of the eigenvalues doesn't really matter. What, that's not obvious to you from the statement?