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aglearner
aglearner
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Suppose we have a real analytic family of matrices $A_t, B_t\in GL_n(\mathbb C)$, with $t\in \mathbb R$. Suppose that for $t\in (0,1)$ there is a common non-zero eigenvector $v_t\in \mathbb C^n\setminus 0$ such that $A_t(v_t)=B_t(v_t)=0$$A_t(v_t)=B_t(v_t)=v_t$. Is it true that such a common (non-zero) vector exists for all $t\in \mathbb R$?

Suppose we have a real analytic family of matrices $A_t, B_t\in GL_n(\mathbb C)$, with $t\in \mathbb R$. Suppose that for $t\in (0,1)$ there is a common non-zero eigenvector $v_t\in \mathbb C^n\setminus 0$ such that $A_t(v_t)=B_t(v_t)=0$. Is it true that such a common (non-zero) vector exists for all $t\in \mathbb R$?

Suppose we have a real analytic family of matrices $A_t, B_t\in GL_n(\mathbb C)$, with $t\in \mathbb R$. Suppose that for $t\in (0,1)$ there is a common non-zero eigenvector $v_t\in \mathbb C^n\setminus 0$ such that $A_t(v_t)=B_t(v_t)=v_t$. Is it true that such a common (non-zero) vector exists for all $t\in \mathbb R$?

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aglearner
aglearner
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  • 40
  • 99

Fixed subspaces of a family of representations $\rho_t: F_2\to GL(n,\mathbb C)$

Suppose we have a real analytic family of matrices $A_t, B_t\in GL_n(\mathbb C)$, with $t\in \mathbb R$. Suppose that for $t\in (0,1)$ there is a common non-zero eigenvector $v_t\in \mathbb C^n\setminus 0$ such that $A_t(v_t)=B_t(v_t)=0$. Is it true that such a common (non-zero) vector exists for all $t\in \mathbb R$?