Meaning of the spectral norm of a matrix

The spectral norm is the maximum singular value of a matrix. Intuitively, you can think of it as the maximum 'scale', by which the matrix can 'stretch' a vector.

Let us consider the singular value decomposition (SVD) of a matrix $X = U S V^T$, where $U$ and $V$ are matrices containing the left and right singular vectors of $X$ in their columns. $S$ is a diagonal matrix containing the singular values. A intuitive way to think of the norm of $X$ is in terms of the norm of the singular value vector in the diagonal of $S$. This is because the singular values measure the energy of the matrix in various principal directions.

One can now extend the $p$-norm for a finite-dimensional vector to a $m\times n$ matrix by working on this singular value vector:

\begin{align} ||X||_p &= {\Big(} \sum_{i=1}^{\text{min}(m,n)} \sigma_i^p {\Big)}^{1/p} \end{align}

This is called the Schatten norm of $X$. Specific choices of $p$ yield commonly used matrix norms:

1. $p=0$: Gives the rank of the matrix (number of non-zero singular values).
2. $p=1$: Gives the nuclear norm (sum of absolute singular values). This is the tightest convex relaxation of the rank.
3. $p=2$: Gives the Frobenius norm (square root of the sum of squares of singular values).
4. $p=\infty$: Gives the spectral norm (max. singular value).