Spectral norm

Definition

For a real matrix

The spectral norm of a $n\times n$ square matrix $A$ with real entries is defined in the following equivalent ways:

1. It is the maximum of the Euclidean norms of vectors $A{\vec {x}}$ where ${\vec {x}}$ is on the unit sphere, i.e., has Euclidean norm 1.
2. It is the maximum, over all nonzero vectors ${\vec {x}}\in \mathbb {R} ^{n}$, of the quotients ${\frac {\|A{\vec {x}}\|}{\|{\vec {x}}\|}}$ where $\|\cdot \|$ denotes the Euclidean norm.
3. It is the largest singular value of $A$, or equivalently, it is the square root of the largest eigenvalue of the product $AA^{T}$.

For a complex matrix

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