matrix norms are bounded below by the spectral radius

[[concept]]

Topics

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Theorem

Suppose $||\cdot ||$ is a matrix norm on $M_n$. Then for all $A\in M_n$, we have $\rho (A)\le ||A||$.

$||A||={\max }_x\frac{||A||}{||x||}={\lambda }_{\max }$ (we simply choose some norm 1 eigenvector of $A$ with the maximum eigenvalue).

If the norm is an induced norm, then the result is trivial:

For any matrix norm, we let $x$ be an eigenvector associated with $\lambda$ an eigenvalue of maximum modulus. Set $B$ to be the $n\times n$ matrix with $x$ as each column. Then $||AB||=||Ax|Ax|\dots |Ax||=||\lambda B||\le ||A||\cdot ||B||$ Since $x\ne 0$ this implies that $B\ne 0⟹||B||\ne 0$. Thus we have that $\rho (A)=|\lambda |\le ||A||$

References

References

See Also

Mentions

Mentions

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