# Math Help - Determining a matrix according to its charateristic polynomial

1. ## Determining a matrix according to its charateristic polynomial

I am wondering whether it is possible to determine a matrix if its characteristic polynomial is known. Of course, there is more than one matrix for a fixed characteristic polynomial but there must also be something special that other matrixes dont have. For instance, if the c.p. is $(-1)^n \left(t^n-1\right)$ with n the order, are there some simple conditions that can be applied to its matrix?

2. The eigenvalues (and so the characteristic polynomial) determine a matrix up to "similarity".

Matrices having the same eigenvalues (characteristic polynomial) must be "similar"- that is, $P= QDQ^{-1}$ for some invertible matrix Q.