The spectral norm of a matrix is its maximum singular value.
Let $A$ be an $n\times n$ matrix. Its spectral norm, also known as the induced norm $||A||_2$, is its maximum singular value, i.e. $$ ||A||_2 = \max_{\mathbf{x}\neq \mathbf{0} }\frac{||A\mathbf{x}||_2}{||\mathbf{x}||_2} $$Alternatively, one could consider $B=A^HA$. Since $B$ is hermitian and positive semi-definite, it has positive real eigenvalues: the spectral norm of $A$ is the square root of the largest of these.
A closely related term is the spectral radius $\rho(A)$, which is the maximum absolute value of the eigenvectors of $A$. These are not necessarily the same: for instance, if $A=\pmatrix{0 & 1\\ 0 & 0}$, then $\rho(A)=0$ and $||A||_2=1$. In general we have $\rho(A)\leq ||A||_k$ and a powerful result known as Gelfand's formula gives $\rho(A) = \lim_{k\to \infty}||A||_k ^{1/k}$. See spectral-radius for these questions.
Questions about the spectral norm should usually contain some combination of the tags linear-algebra, eigenvalues-eigenvectors, matrices, svd, or similar.
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