matrix norms are bounded below by the spectral radius

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Theorem

Suppose $| | \cdot | |$ is a matrix norm on $M_{n}$ . Then for all $A \in M_{n}$ , we have $ρ (A) \leq | | A | |$ .

Proof

If the norm is an induced norm, then the result is trivial:
$| | A | | = max_{x} \frac{| | A | |}{| | x | |} = λ_{max}$
(we simply choose some norm 1 eigenvector of $A$ with the maximum eigenvalue).

For any matrix norm, we let $x$ be an eigenvector associated with $λ$ an eigenvalue of maximum modulus. Set $B$ to be the $n \times n$ matrix with $x$ as each column. Then

| | A B | | = | | A x | A x | \dots | A x | | = | | λ B | | \leq | | A | | \cdot | | B | |

Since $x \neq 0$ this implies that $B \neq 0 ⟹ | | B | | \neq 0$ . Thus we have that

ρ (A) = | λ | \leq | | A | |

References

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matrix norms are bounded below by the spectral radius

References

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