Spectral norm

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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 \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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