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If A has eigenvector $\mathbf{v}_1$ so that $A\mathbf{v}_1=\lambda_1\mathbf{v}_1$and B has eignenvector $\mathbf{v}_2$ so that $B\mathbf{v}_2=\lambda_2\mathbf{v}_2$, then what can you say about AB? can you say $AB\mathbf{v_3}=\lambda_3\mathbf{v}_3$? and what would be the relationship between $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ and what would be the relationship between $\lambda_1,\lambda_2,\lambda_3$?

Edit $A,B$ are 3 by 3 matrices and $\lambda_1,\lambda_2,\lambda_3$ can be real or complex numbers and $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ is a triple.

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    Any (real) $3\times 3$ matrix will have eigenvectors, so $AB$ certainly has eigenvectors. But there need not be any relation between $\mathbf{v}_2$ and $\mathbf{v}_3$, or between $\mathbf{v}_1$ and $\mathbf{v}_3$ (although there *can* be relations between them, depending on the specific $A$ and $B$, or on the choice of $\mathbf{v}_2$). I find this question somewhat confusing.2011-03-17

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