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Linear algebra: Similar matrices?
Let A be a complex square upper triangular matrix of rank r. Moreover, the elements of the main diameter are only 0 or 1. Can we always find an upper triangular matrix, similar to A such that the first r elements of the main diameter are 1 and the others are zero? If not, please give a counterexample.
Thank you.
Thank you David. What if the scalar 1 appears exactly r times in the main diameter of A? Indeed if both have the same characteristic polynomial.
1 Answer
- DavidLv 71 decade agoFavorite Answer
if two matrices are similar, they must have the same reduced row echelon form. now consider the matrix A =
[0 1]
[0 0]
and B =
[1 0]
[0 0]
these are both upper triangular with only 1's and 0's on the main diagonal, and are both of rank 1. but they are also both in rref, and so A is not similar to B.
