I saw below theorem in my test book(by friedberg)
Theorem : a linear operator $T$ on $n$-dimensional vector space $V$ over field $K$is diagonalizable "if and only if" both of the condition holds.
1) the characteristic polynomial of $T$ splits over $K$
2) algebraic multiplicity of each eigenvalue is equal to geometric multiplicity.
While in other textbook I saw theorem that,
Theorem: a linear operator $T$ on $n$-dimensional vector space $V$ over field $K$is diagonalizable "if and only if" condition (2) of above theorem holds.
So which of these two theorems are true? First one or second one?
Further,
If 'first one' is true then can anyone given me example of linear operator $T$ on $n$-dimensional space which is not diagonalizable but satisfies condition (2). Thank you.
The second theorem is certainly false, any real matrix that does not have real eigenvalues is a counterexample.
If you add the condition
(3) ... and the sum of the geometric multiplicities of the eigenvalues is equal to the dimension of the space
then diagonalizability is equivalent to (2)(3).