I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
2026-03-26 17:37:13.1774546633
Is the spectral radius of a matrix a convex norm of it?
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The spectral radius is the biggest eigenvalue of a matrix.
There is a linked norm called the spectral norm, which is infact the square root of the biggest eigenvalue of the matrix $A^*A$. (So not linked to its own spectral radius, but to the spectral radius of $A^*A$)
Norms are always convex. Due to triangle inequality and to the homogeneity.
Edit:
Biggest eigenvalue is of course not completly correct. You have to consider the absolute values of the eigenvalues...