Let \(A\) be an \(N\times N\) matrix. Let \(p(A)=\sup_{n\geq 0}\| A^ n\|\) and \(r(A)=\sup_{| z| >1}(| z| -1)\| (zI- A)^{-1}\|\), where \(\| \cdot \|\) is the Euclidean norm. If \(p(A)<\infty\) then \(r(A)<\infty\). \textit{H.-O. Kreiss} [ibid. 2, 153-181 (1962; Zbl 0109.34702)] proved that \(r(A)<\infty\) implies \(p(A)<\infty\). \textit{E. Tadmor} [Linear Algebra Appl. 41, 151-159 (1981; Zbl 0469.15011)] proved that \(p(A)\leq (32e/\pi)N r(A)\) and conjectured that the best bound is linear in \(N\). The authors of the present paper verify this conjecture by exhibiting a family \(\{A_ N\}\) of matrices with \(p(A_ N)\sim e Nr(A_ N)\) as \(N\to \infty\). They improve Tadmor's bound to \(p(A)\leq 2eN r(A)\) and conjecture that the 2 is unnecessary.