I'm reading Hoffman and Kunze's linear algebra and on page 313 they prove the following theorem:
Theorem 16 On a finite-dimensional inner product space of positive dimension, every self-adjoint operator has a (non-zero) characteristic vector.
They proved this in the following way:
I didn't understand:
- where do they use the fact $A=A^*$
- Why does $c$ be a real scalar matters?
To sum up: Why don't they simply say the characteristic polynomial, $\det(xI-A)$, is a polynomial of degree $n$ over the complex numbers then there is $c$ such that $\det (cI-A)=0$ and then there is a non-zero $X$ such that $AX=cX$ which follows there is a non-zero vector $\alpha\in V$ such that $T\alpha=c\alpha$?