Let the matrix $A$ have distinct eigenvalues. Then we know that matrix is diagonalizable and the eigenvector forms a basis and $A = V D V^{-1}$, where columns of $V$ are eigenvector of $A$. If we do the Gram-Schmidt process on the basis we can get an orthonormal basis and also the columns of $V$ can be changed to that. Then if we take infinite norm of the matrix $A$, it is same as largest absolute eigenvalue of $A$. Is my reasoning correct?
By infinite norm, I mean
$$\| A \|_{\infty} := \max_{i}\sum_{j}\vert a_{ij}\vert$$
Is the statement "the infinite norm of the diagonalizable matrix $A$ is the same as largest absolute eigenvalue of $A$" correct?