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Here is a theorem about convergence of iterative methods for linear system in Burden and Faires' book "Numerical Analysis"

For any $x_0 \in \mathbb{R}^n$, the sequence defined by $x^k = Tx_{k-1} + c$ converges to the unique solution of $x = Tx + c$ if and only if the spectral radius of T is less than 1.

I understand that the iteration converges to a solution of $x = Tx + c$, but how do we know that a solution to $x = Tx + c$ is unique?

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2 Answers 2

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The solution is not unique in general. For the uniqueness of the solution we need $I - T$ is nonsingular. If we have $\rho(T) < 1$, then the eigenvalues of $I - T$ are $1 - \lambda_i$ which has modulus strictly greater than $1 - \rho(T) > 0$. Therefore $I - T$ is nonsingular.

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  • $\begingroup$ Is is a theorem that if $\lambda_i$ are the eigenvalues of $T$, then the eigenvalues of $I - T$ are $1 - \lambda_i$? $\endgroup$
    – user74261
    Commented Apr 29, 2015 at 4:22
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    $\begingroup$ I would rather say it is a proposition because the proof is straightforward. $\lambda$ is an eigenvalue for $T$ if and only if $\det(T - \lambda I) = 0$ if and only if $\det((I-T) - (1 - \lambda)I) = 0$ if and only if $1 - \lambda$ is an eigenvalue for $I-T$. $\endgroup$
    – Empiricist
    Commented Apr 29, 2015 at 4:27
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If $ρ(T)<1$ then there exists a norm on the vector space such that in its operator norm $\|T\|_{op}<1$. The statement of convergence and uniqueness then follows from Banach's fixed-point theorem.

See also the (Carl) Neumann operator power series.

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