Math Help - Algebra, Problems For Fun (19)

1. Algebra, Problems For Fun (19)

A standard notation: By $M_n(R)$ we mean the set of all $n \times n$ matrices with entries from some set $R.$

Problem: Let $A \in M_n(\mathbb{C}).$ Suppose that $\text{tr}(A)=\text{tr}(A^2)=\cdots = \text{tr}(A^n)=0.$ Prove that $A^n=\bold{0}.$

Remark: The above result remains true if we replace $\mathbb{C}$ with any field $F$ of characteristic 0. It is not necessarily true if $\text{char} F \neq 0.$ A counter-example is: $A=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \in M_2(\mathbb{F}_2).$

2. Using the fact that the characteristic polynomial of an nxn matrix is a monic polynomial with coefficients that can be expressed in terms of it's trace and the traces of it's powers upto the power n we get that

$\chi_A(X) = \pm X^n$ as all the traces are 0.

Then the result follows from the Cayley Hamilton Theorem.

I couldn't think of a way of proving it that didnt rely on these two results but I'm sure there is a nice way.

thanks, pomp.

3. Originally Posted by pomp
Using the fact that the characteristic polynomial of an nxn matrix is a monic polynomial with coefficients that can be expressed in terms of it's trace and the traces of it's powers upto the power n we get that

$\chi_A(X) = \pm X^n$ as all the traces are 0.

Then the result follows from the Cayley Hamilton Theorem.

I couldn't think of a way of proving it that didnt rely on these two results but I'm sure there is a nice way.

thanks, pomp.
let $\lambda_1, \cdots , \lambda_n$ be the eigenvalues of $A.$ we know that every matrix over $\mathbb{C}$ is similar to some triangular matrix. so $A=PBP^{-1},$ for some triangular matrix $B.$

then $0=\text{tr}(A^k)=\text{tr}(PB^kP^{-1})=\text{tr}(B^k)=\lambda_1^k + \cdots + \lambda_n^k,$ for all $1 \leq k \leq n,$ which gives us $\lambda_1=\cdots=\lambda_n=0.$ therefore $A^n=\bold{0},$ by Cayley-Hamilton.