Spectral norm and inner product

Question

We know that for a general $N\times n$ matrix $A$, its spectral norm is defined as

$$\|A\|=\sup_{x\in S^{n-1}} \|Ax\|_{2},$$

where $x$ is from the unit sphere.

My question is that why when $A$ is a $n\times n$ symmetric matrix, $\|A\|$ can also be defined as

$$\|A\|=\sup_{x\in S^{n-1}} |\langle Ax,x\rangle|,$$

and why this does not imply

$$\|Ax\|_{2}=|\langle Ax,x\rangle|?$$

Answer 1

$\|Ax||_2=|\langle Ax, x \rangle|$ cannot be true for a simple reason. Change $x$ to $cx$ and see what happens!

Answer 2

Hint: A symmetric matrix is diagonalizable over an orthonormal basis, and thus both values coincide with $\max_i|\lambda_i|$ where $\lambda_i$ are the eigenvalues.