Vipul: Created page with "==Definition== ===For a real matrix=== The '''spectral norm''' of a $n \times n$ square matrix $A$ with real entries is defined in the following equiva..."

2014-05-09T16:44:07Z

Created page with "==Definition== ===For a real matrix=== The '''spectral norm''' of a <math>n \times n</math> square matrix <math>A</math> with real entries is defined in the following equiva..."

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==Definition==

===For a real matrix===

The '''spectral norm''' of a <math>n \times n</math> square matrix <math>A</math> with real entries is defined in the following equivalent ways:

# It is the maximum of the [[Euclidean norm]]s of vectors <math>A\vec{x}</math> where <math>\vec{x}</math> is on the unit sphere, i.e., has Euclidean norm 1.
# It is the maximum, over all nonzero vectors <math>\vec{x} \in \R^n</math>, of the quotients <math>\frac{\| A\vec{x} \|}{\| \vec{x} \|}</math> where <math>\| \cdot \|</math> denotes the Euclidean norm.
# It is the largest singular value of <math>A</math>, or equivalently, it is the square root of the largest eigenvalue of the product <math>AA^T</math>.

===For a complex matrix===

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Spectral norm - Revision history

Vipul: Created page with "==Definition== ===For a real matrix=== The '''spectral norm''' of a n \times n square matrix A with real entries is defined in the following equiva..."

Vipul: Created page with "==Definition== ===For a real matrix=== The '''spectral norm''' of a $n \times n$ square matrix $A$ with real entries is defined in the following equiva..."