So while studying for my Linear Algebra test, I'm required to study some theorems and their proofs, and I have trouble understanding a particular part of the proof for the following (I'm translating from another language, so I'm sorry for any mistakes):
The matrix $A\in M_n(\Bbb{F})$ is diagonalizable if and only if the following two conditions are met:
- All of the matrix's Eigenvalues are from $\Bbb{F}$
- All of the matrix's Eigenvalues's algebraic multiplicities are equal to their respective geometric multiplicities.
I've also seen the first condition written as: "All of the characteristic polynomial's roots are in $\Bbb{F}$".
And while I have the proof for the 2nd condition pretty much pegged down, I fail to understand the first condition or its proof, which reads (again, crudely translated):
The first condition is equivalent to saying that the sum of the algebraic multiplications is equal to $n$.
I fail to see how. In another source, I found it as:
The first condition is equivalent to saying that the characteristic polynomial of A is a product of polynomials with a maximum rank of $1$.
And finally, this (directly from my professor's notes, and surprisingly - or not - the least helpful):
"If $A$ is diagonalizable then all of its Eigenvalues belong to $\Bbb{F}$.
Very helpful. I doubt I'll get any points if I use that one, though.
So while I see how the first and second sources are similar, I still don't see what the multiplication (or - similarly - the polynomial rank) have to do with the Eigenvalues being from $\Bbb{F}$.
Am I missing something basic? I'm I simply misunderstanding the definition? Thanks in advance for any assistance understanding this theorem.