Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 1 down vote accepted

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

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.