Chapter 5.2, Problem 95E

Guided proof Let $\langle u,v\rangle$ be the Euclidean inner product on $R^n$. Use the fact that $\langle u,v\rangle =u^{T}v$ to prove that for any $n\times n$ matrix $A$,

(a) $\langle A^{T}Au,v\rangle =\langle u,Av\rangle$

and

(b) $\langle A^{T}Au,u\rangle ={\left|\left|Au\right|\right|}^2$

Getting started: To prove (a) and (b), make use of both the properties of transposes (Theorem 2.6) and the properties of the dot product (Theorem 5.3).

(i) To prove part (a), make repeated use of the property $\langle u,v\rangle =u^{T}v$ and Property 4 of Theorem 2.6.

(ii) To prove part (b), make use of the property $\langle u,v\rangle =u^{T}v$, Property 4 of Theorem 2.6, and Property 4 of Theorem 5.3.