query: The characteristic polynomial of $A$ is $x^n$ if and only if $\text{Tr}(A^i)=0$ for all $1\le i \le n$.
positive_passages.docid: 1973
positive_passages.text: Traces of all positive powers of a matrix are zero implies it is nilpotent
negative_passages.docid: 19280
negative_passages.text: Jordan Canonical Form determined by characteristic and minimal polynomials in dimension $3$, but not beyond
negative_passages.docid: 76694
negative_passages.text: Proving that $UT=TU$ iff $U=g(T)$ for some polynomial $g$
negative_passages.docid: 149364
negative_passages.text: Characteristic polynomial of an inverse
negative_passages.docid: 114812
negative_passages.text: Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+...+a_0$ be a complex polynomial satisfying $|p(z)| \leq 1$ for $|z|\leq 1$, then prove that $p(z)=z^n$
negative_passages.docid: 52528
negative_passages.text: Showing that minimal polynomial has the same irreducible factors as characteristic polynomial
negative_passages.docid: 65224
negative_passages.text: Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
negative_passages.docid: 83337
negative_passages.text: If $A ∈M_n(C)$ is such that $(Ax,x) = 0$ for all $x ∈C^n$, then $A = 0$
negative_passages.docid: 14836
negative_passages.text: A polynomial sequence
negative_passages.docid: 67058
negative_passages.text: Prove that $\text{rank}(X^TX)=\text{rank}(X)$
negative_passages.docid: 106585
negative_passages.text: Show $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$.
negative_passages.docid: 102464
negative_passages.text: Prove $R < ( 1 + (\sum |a_{i}|^{p} )^{q/p})^{1/q}$ for absolute value of polynomial zeros
negative_passages.docid: 46355
negative_passages.text: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
negative_passages.docid: 97729
negative_passages.text: Show that a polynomial $f(x)$ over a field $k$ is irreducible if and only if the polynomial $f(x+1)$ is irreducible
negative_passages.docid: 76955
negative_passages.text: Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros
negative_passages.docid: 81063
negative_passages.text: If $p$ is a polynomial, then either $\,p(z)=z^n\,$ or $\,\max_{|z|=1}|p(z)|>1$.
negative_passages.docid: 128043
negative_passages.text: $A^n=0$ for $A$ nilpotent without the use of minimal polynomial