Return to Answer

Re your answer:

Statement 1 follows from Weyl's inequalities and $\rho_1(C) = O(\lambda^{-1})$.

$2 \implies 3$ seems problematic; what if $\rho_i(M)=0$ for $i>1$, which is perfectly possible given the definition of big-O? For instance, $C = \frac{1}{\lambda^3}I $ should give $\rho_1(M^{-1}) \sim \lambda^2 \not\in O(\lambda)$.

Statement 4 seems just the matrix norm inequality $\|M^{-1}e\| \leq \|M^{-1}\|\|e\|$.

Re your answer:

Statement 1 follows from Weyl's inequalities and $\rho_1(C) = O(\lambda^{-1})$.

$2 \implies 3$ seems problematic; what if $C = \frac{1}{\lambda^3}I$, which should be possible given the definition of big-O? Then $\rho_1(M^{-1}) \sim \lambda^2 \not\in O(\lambda)$.

Statement 4 seems just the matrix norm inequality $\|M^{-1}e\| \leq \|M^{-1}\|\|e\|$.

Re your answer:

Statement 1 follows from Weyl's inequalities and $\rho_1(C) = O(\lambda^{-1})$.

$2 \implies 3$ seems problematic; what if $\rho_i(M)=0$ for $i>1$, which is perfectly possible given the definition of big-O? For instance, $C = (-\frac{1}{\lambda^2}+ \frac{1}{\lambda^3})I $ should give $\lambda_1(M^{-1}) = \lambda^3 \not\in O(\lambda)$.

Statement 4 seems just the matrix norm inequality $\|M^{-1}e\| \leq \|M^{-1}\|\|e\|$.

Re your answer:

Statement 1 follows from Weyl's inequalities and $\rho_1(C) = O(\lambda^{-1})$.

$2 \implies 3$ seems problematic; what if $\rho_i(M)=0$ for $i>1$, which is perfectly possible given the definition of big-O? For instance, $C = \frac{1}{\lambda^3}I $ should give $\rho_1(M^{-1}) \sim \lambda^2 \not\in O(\lambda)$.

Statement 4 seems just the matrix norm inequality $\|M^{-1}e\| \leq \|M^{-1}\|\|e\|$.

Re your answer:

Statement 1 follows from Weyl's inequalities and $\rho_1(C) = O(\lambda^{-1})$.

$2 \implies 3$ seems problematic; what if $\rho_i(M)=0$ for $i>1$, which is perfectly possible given the definition of big-O? For instance, $C = \frac{1}{\lambda^2}I$.

Statement 4 seems just the matrix norm inequality $\|M^{-1}e\| \leq \|M^{-1}\|\|e\|$.

Re your answer:

Statement 1 follows from Weyl's inequalities and $\rho_1(C) = O(\lambda^{-1})$.

$2 \implies 3$ seems problematic; what if $\rho_i(M)=0$ for $i>1$, which is perfectly possible given the definition of big-O? For instance, $C = (-\frac{1}{\lambda^2}+ \frac{1}{\lambda^3})I $ should give $\lambda_1(M^{-1}) = \lambda^3 \not\in O(\lambda)$.

Statement 4 seems just the matrix norm inequality $\|M^{-1}e\| \leq \|M^{-1}\|\|e\|$.