What is the difference between "the distinct eigenvalues" and "distinct eigenvalues"? I always get help from here so I very appreciate it.
I'm studying linear algebra,specifically digonalizability part.
and I have a question about this Theorem.
'Theorem 5.9 Let T be a linear operator on a finite dimensional vector space V such that
the characteristic polynomial of T splits. Let lamda(1), lamda(2), ...lamda(k) be the distinct eigenvalues of T. Then ~'
in this paragraph , are (lamda(1), lamda(2), ...lamda(k)) the all eigenvalues for operator T?
or is there a possibility that there are more eigenvalues than (lamda(1), lamda(2), ...lamda(k))?
this paragraph is from 'Linear algebra, friedberg'
I not good at english yet, so I can't tell apart the details.
in this book, some theorem just say ' let ~~be distinct eigenvalues of T ' without 'the'
 A: This is a subtle point in English.
Suppose I have a transformation $T$ with eigenvalues $2,2, 3, 5$, and I say "let $b$ and $c$ be distinct eigenvalues of $T$". That means that $b$ and $c$ are eigenvalues (hence must be $2, 3,$ or $5$) and must be distinct (i.e., they cannot both be $2$). If $T$ had eigenvalues $1,2,3$, the same rule will still apply. If you said "let $b$ and $c$ be eigenvalues of $T$," they could both be $3$, because they would both be eigenvalues of $T$. Saying "distinct" rules out this possibility.
In your theorem, you know that the characteristic polynomial of $T$ splits; so it's a product of factors $(x-a_1)^{n_1} \cdots (x- a_k)^{n_k}$. Let's look at a concrete example: suppose the polynomial is $(x-2)^2 (x-3) (x-5)^2$. Then one listing of the eigenvalues would be
2, 2, 3, 5, 5
but these numbers are not distinct. When the author says "the distinct eigenvalues", what's meant is "2, 3, and 5". Because (1) they have to be eigenvalues, (2) they have to be distinct, and (3) they have to be all the possible distinct eigenvalues. That third point is the subtle one, because the word "all" doesn't appear anywhere!
When we say "OK, folks, gather for a photo. I want the tall people in the back!", we mean "all people who are tall should stand at the back". The use of "the" in this context somehow (for grammar reasons I don't understand) means "all".
I hope this helps a bit.
