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Let $A$ be an arbitrary square matrix and define $ \rho(A)$ to be the maximal eigenvalue of $A$ in absolute value. If $ \rho(A)<1,$ then there exists a norm of $A$ such that $ \| A \|<1.$ How to do this? Thanks.

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2 Answers

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We use the following result: $\rho(A)=\inf\{\lVert A\rVert,\lVert\cdot\rVert\mbox{ matrix norm}\}$ and apply the definition of $\inf$ to $\varepsilon:=\frac{1-\rho(A)}2$.

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If $A$ is symmetric then the Euclidean norm of $A$ is $\rho(A)$. If it is not symmetric, define the block matrix $$ B=\begin{array} 0 0 & A \\ A^t & 0 \end{array} \ . $$ Then the norm of $A$ is equal to norm of $B$ and you may define the norm of $A$ that way.

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