Question

3. Let V be a finite dimensional vector space with a positive definite scalar product. Let A: V-> V be a symmetric linear map. We say that A is positive definite if (Av, v) > 0 for all ve V and v 0. Prove: (a) if A is positive definite, then all eigenvalues are > 0. (b) If A is positive definite, then there exists a symmetric linear map B such that B2 = A and BA = AB. What are the eigenvalues of B? [Hint: Use a basis of V consisting of eigenvectors.]

Answer 1

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the NEV:A en ace Sime e so Bis well ditmed Let Now -S if +s K. *Then BェB.ら = So A 2 Noa AB = ZAE Pe BA 1 1 use Chus BA AB in by Speckal theaglon v Sine B= eemplet liat of diotine VA, V2, k ge the egen values of B.