Exercise 6.5.14

Answers

We may write $A = P^{∗} BP$ . Compute

⟨ L_{A} (x), x ⟩ = ⟨ L_{P^{∗} BP} (x), x ⟩

= ⟨ L_{P}^{∗} L_{B} L_{P} (x), x ⟩ = ⟨ L_{B} (L_{P} (x)), L_{P} (x) ⟩ .

If $A$ is positive definite, for each vector $y$ we may find some $x$ such that $L_{P} (x) = y$ since $P$ is invertible. Also, we have

⟨ L_{B} (y), y ⟩ = ⟨ L_{B} (L_{P} (x)), L_{P} (x) ⟩ = ⟨ L_{A} (x), x ⟩ > 0 .

If $B$ is positive definite, we may check that

⟨ L_{A} (x), x ⟩ ⟨ L_{B} (L_{P} (x)), L_{P} (x) ⟩ > 0 .

jephianlin

2011-06-27 00:00