The p-norm of some matrices

Abstract

We compute the operator $p$-norm of some $n\times n$ complex matrices, which can be seen as bounded linear operators on the $n$ dimensional Banach space ${\ell }^p(n)$. The notion of logarithmic affine matrices is defined, and for such a matrix its $p$-norm is computed exactly. In particular, a matrix $A=\left(\begin{array}{ccc}8& 1& 6\\ 3& 5& 7\\ 4& 9& 2\end{array}\right)$ which corresponds to a magic square belongs to the class of logarithmic affine matrices, and its $p$-norm is equal to $15$ for any $p\in [1,\infty ]$.

18 pages