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Operator norm

{title2=$\|A\|_2$}

= Spectral norm
{synonym}

The matrix 2-norm is the <operator norm> induced by the <Euclidean norm>:
$$
\|A\|_2=\sup_{x\ne0}\frac{\|Ax\|_2}{\|x\|_2}.
$$
It equals the largest <singular value> of $A$ and is invariant under multiplication by <orthogonal matrix>[orthogonal] or <unitary operator>[unitary] matrices.

The <matrix norm> induced by Euclidean vector norms is $\|A\|_2=\sup_{x\ne0}\|Ax\|_2/\|x\|_2$, equal to the largest singular value.

Matrix 2-norm

Matrix 2-norm of a normal matrix

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 36 / 2 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 36 / 2 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 4 / a / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 40E / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 41E / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 339 / 2 / d / Solution

codex/spectral-norm

Matrix 2-norm ( $∥ A ∥_{2}$ )

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Matrix 2-norm (∥A∥2​)