I’m having trouble understanding how Lang in his third edition of Linear Algebra “reformulated” theorem 3.1 into 4.1.
This is theorem 3.1, which I totally understand:
Let $A$ be a symmetric $n$ by $n$ real matrix. Then there exists a nonzero real eigenvector for $A$.
This is theorem 4.1, which was said to have been reformulated from the above:
Let $V$ be a finite dimensional vector space with a positive definite symmetric bilinear form. Let $A:V\to V$ be a symmetric linear map. Then $A$ has a nonzero eigenvector.
I don’t know how 4.1 is related to 3.1 and I’ve been trying to figure this out for some time. Any help would be appreciated.