matrix norm

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Matrix Norm

A norm $| | \cdot | |$ on $M_{n} (K)$ over $K$ is a matrix norm if for all $A, B \in M_{n}$ we have

| | A B | | \leq | | A | | \cdot | | B | |

(this condition is called submultiplicativity)

If, in particular, $| | \cdot | |^{'}$ is any norm on $K^{n}$ and matrices in $M_{n}$ are $K^{n}, | | \cdot | |^{'} \to K^{n}, | | \cdot | |^{'}$ , then the operator norm induced by that vector norm $| | \cdot | |_{| | \cdot | |^{'}, | | \cdot | |^{'}}$ is an induced matrix norm. (We have already shown this)

Example

$| | A | |_{2, 2}$ , $| | A | |_{1, 1}$ (maximum absolute column sum), $| | A | |_{\infty, \infty}$ (maximum absolute row sum)

Where did we already show this? We showed that $T : U \to V, S : V \to W$ satisfies

| | S \circ T | | \leq | | S | | \cdot | | T | |

Exercise

$| | \cdot | |_{1}$ is a matrix norm ie NTS $| | A B | |_{1} \leq | | A | |_{1} \cdot | | B | |_{1}$
$| | \cdot | |_{F}$ is a matrix norm (frobenius norm)
$| | \cdot | |_{\infty}$ is NOT a matrix norm

counterexample for (3)

Consider $A = B =$ matrix of all 1s. Then $A B$ is the matrix of all 2s.

Relationship between norms on square matrices

induced matrix norms $\subset$ matrix norms $\subset$ norm on $M_{n} (K)$

If we can prove convergence for general norm on matrices, then we also have convergence for matrix norms and induced matrix norms

References

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matrix norm

References

See Also

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