What is the ratio between $|\langle Av,v \rangle |$ and $\|v\|^2$ where $A$ is an $n\times n$ unitary matrix and $v\in \mathbb C^n$? I'm trying to determine the ratio between $|\langle Av,v \rangle |$ and $\|v\|^2$ where $A$ is an $n\times n$ unitary matrix and $v\in \mathbb C^n$. In particular I'm trying to determine whether $|\langle Av,v \rangle | \le \|v\|^2$ or $|\langle Av,v \rangle | \gt \|v\|^2$.
This is what I have so far:
Let $w,u\in \mathbb C^n$ such that $w+u=v$ then:
Computing $|\langle Av,v \rangle |$ :
$$|\langle Av,v \rangle | =|\langle A(w+u),w+u \rangle |=|\langle Aw+Au,w+u \rangle | = |\langle Aw,w \rangle + \langle Aw,u \rangle + \langle Au,w \rangle + \langle Au,u \rangle |$$
Computing $\|v\|^2$ :
$$\|v\|^2 = \langle v,v \rangle = \langle w+u,w+u \rangle = \langle w,w \rangle + \langle w,u \rangle + \langle u,w \rangle + \langle u,u \rangle$$
I'm not quite sure how to continue. I can use the exchange lemma and the fact that $A^{-1} = A^*$ because $A$ is an unitary matrix but I can't see how it helps me to simply the first equation.
Note : $\langle \cdot \rangle$ denotes the dot product.
 A: Hint:
A very famous theorem allows us to state (fill in the blanks) 
$$
\left|\langle Av, v\rangle\right|\leqslant\left\|Av\right\|\left\|v\right\|\leqslant \left\|A\right\|\ldots\,,
$$
And, don't forget that $A$ is unitary.
A: $$|\langle Av,v\rangle|=|v^*A^Hv|\le_{CS} \|Av\|_2\|v\|_2=\|v\|_2^2,\ (\because\|Av\|_2^2=\|v\|_2^2\quad\mbox{by unitarity of }A)$$ CS is Cauchy-Scwartz inequality.
A: You can do better than Cauchy Schwarz by observing that you are trying to maximize (i.e. find the maximum attained value) a quadratic form. Indeed, you are given the function
$$
Q(v)=\langle Av, v\rangle$$
which is homogeneous of degree 2, that is 
$$
Q(zv)=\lvert z\rvert^2Q(v), \qquad \forall z\in \mathbb{C},\ \forall v\in \mathbb{C}^n,$$
and you are trying to find 
$$\tag{1}M=\max\left\{ |Q(v)|\, :\, v\in \mathbb{C}^n\ \lVert v\rVert =1\right\}.$$ 
(Homogeneity then gives the sharp inequality $|Q(v)|\le M\lVert v\rVert^2,\ v\in \mathbb{C}^n$). To solve the maximization problem (1) you can use the spectral theorem to unitarily diagonalize $A$: 
$$
A=U\,\mathrm{diag}(\lambda_1\ldots\lambda_n)U^\star, $$
where $U$ is unitary. Therefore 
$$
|Q(v)|=\left\lvert\sum_{j=1}^n \lambda_j \lvert w_j\rvert^2\right\rvert, \qquad w=U^\star v.$$
Since $U$ is unitary one has $\lVert w\rVert=\lVert v\rVert=1$. Moreover, $A$ is unitary too, which implies that all of its eigenvalues have modulus $1$:  $\lvert\lambda_j\rvert=1$. We conclude that 
$$
|Q(v)|\le \sum_{j=1}^n \lvert\lambda_j\rvert \lvert w_j\rvert^2 = \sum_{j=1}^n \lvert w_j\rvert^2=1.
$$
Therefore $M\le 1$. Evaluating $Q$ at $w=(1, 0, \ldots , 0)$ we see that actually $M=1$. 
The advantage of this method over Cauchy Schwarz is that it not only gives a bound, it also shows that the bound is sharp. Moreover, the method is flexible and works for any kind of quadratic functions, even in infinite dimensions. 
