# If M is a linear operator on bbbR^3 with unique and real eigenvalues lambda_1<lambda_2<lambda_3, such that EE x in bbbR^3 \\{0}, satisfying the condition lim_(n to oo)norm(M^n x)=0. What are the possible values of lambda_1?

If M is a linear operator on ${\mathbb{R}}^{3}$ with unique and real eigenvalues ${\lambda }_{1}<{\lambda }_{2}<{\lambda }_{3}$, such that $\mathrm{\exists }x\in {\mathbb{R}}^{3}\setminus \left\{0\right\}$, satisfying the condition $\underset{n\to \mathrm{\infty }}{lim}||{M}^{n}x||=0$. What are the possible values of ${\lambda }_{1}$?
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Lamar Esparza
That's not as straightforward of an answer as it seems.
If x is a linear multiple of ${\zeta }_{1}$, then we can affirm that $|{\lambda }_{1}|<1$
Otherwise, we have $x={c}_{1}{\zeta }_{1}+{c}_{2}{\zeta }_{2}+{c}_{3}{\zeta }_{3}$ and that means ${M}^{n}x={c}_{1}{\lambda }_{1}^{n}{\zeta }_{2}+{c}_{2}{\lambda }_{2}^{n}{\zeta }_{2}+{c}_{3}{\lambda }_{3}^{n}{\zeta }_{3}$, implying all ${\lambda }_{i}$ should have magnitude less than 1 to satisfy that property for any arbitrary vector x.