For a real matrix
The spectral norm of a square matrix with real entries is defined in the following equivalent ways:
- It is the maximum of the Euclidean norms of vectors where is on the unit sphere, i.e., has Euclidean norm 1.
- It is the maximum, over all nonzero vectors , of the quotients where denotes the Euclidean norm.
- It is the largest singular value of , or equivalently, it is the square root of the largest eigenvalue of the product .
For a complex matrix
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