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Questions about rank and eigenvalues of a matrix

Let $A=\begin{pmatrix}1&w&w^2\\w&w^2&1\\w^2&w&1\end{pmatrix}$ Where $w$ is a complex no. s.t. $w^3=1$. Its clear by adding columns of matrix that $0$ is an eigen value of $A$.

Do there exist linearly independent vectors $u,v\in\mathbb{C}^3$ s.t. $Au=Av=0$?

can anyone help me please....

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    I want to find if there exist linearly independent vectors u,v∈C3 s.t. Au=Av=0? – 2012-12-17

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You just want to what is the rank of $A$. Notice that the second and third column are just the first column scaled by $w$ and $w^2$ respectively. Thus, the rank is at most $1$ (so it's precisely $1$, since $A$ is not the zero matrix). In other words, the dimension of the kernel of $A$ is $2$, i.e. there are two linearly independent vectors $u,v$ with $Au = Av = 0$. You can even find them explicitly: one is $(1,1,1)$, another is $(1,w,w^2)$ (other choices are of course possible).

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    @Feanor My fault: when I TeX-ified the OP's question, I wrongly typed the last row of the matrix as $(w^2,1,w)$. It should be $(w^2,w,1)$. – 2012-12-17