Is it true that for jordan block with zero eigenvalue we can choose basis where all diagonal elements are non zero? if there is a proper number 0, then you can try to find a matrix in the form of J^(-1) AJ (A is a matrix that has 0 on the diagonal and all 1 above it), so that it does not have zeros on the diagonal. I tried to write the inverse matrix through the matrix of algebraic complements, but nothing came of it.
2026-03-29 21:53:09.1774821189
Is it true that for jordan block with zero eigenvalue we can choose basis where all diagonal elements are non zero?
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Yes it is true, you have to find a nilpotent matrix represented in another base. For example the 2x2 matrix with the first row equal to (1,1) and the second equal (-1,-1). Call A this matrix and you get that A^2=0 , so its eigenvalues are all 0 and the jordan form is with all zero in the diagonal and one 1. Obviously is not always possible.