Intuitively, the matrix 2-norm measures the greatest action achievable by applying A to x. A = UΣV*, where U and V are orthogonal matrices and Σ is a non-negative diagonal matrix. This is the full singular value decomposition of A. If A is hermitian, then U = V. Basically, A is equivalent to an orthonormal basis change (U) followed by scaling (Σ) and another orthonormal basis change (V). The L^2 norm of a vector is roughly equivalent to the "energy" of a discrete signal in a signal processing sense:
L^2 norm of vector: (xTx)1/2
Energy of discrete signal: xTx
Orthogonal transforms, like rotations and reflections, preserve norms by Parseval's theorem:
(Qx)*(Qx)=x*Q*Qx = x*x
This is why the induced matrix 2-norm of matrix A is equal to its biggest singular value. The matrix 2-norm measures the greatest action achievable by applying A:
||A||2 = supx∈Cn (||Ax||2)/(||x||2)
Basically, the matrix 2-norm measures the greatest action achievable by applying A. Because U and V preserve the L^2 norm, the supremum is achieved, when all of a discrete signal/ vector's energy is scaled by the largest singular value.
||A||2 = supx∈Cn, ||x||2=1 (||Ax||2)
||A||2 = supx∈Cn, ||x||2=1 (||Σx||2)
The unit vector that achieves this largest scaling is the right singular vector v1, or the first column of V.
Notice, that V*v1 = e1, and Σe1 = σmax(A), which is the largest singular value of A.