matrix norm

[[concept]]

Topics

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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 $||AB||\le ||A||\cdot ||B||$ (this condition is called submultiplicativity)

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

$||A|{|}_{2,2}$, $||A|{|}_{1,1}$ (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 $||S\circ T||\le ||S||\cdot ||T||$

Exercise

1. $||\cdot |{|}_1$ is a matrix norm ie NTS $||AB|{|}_1\le ||A|{|}_1\cdot ||B|{|}_1$

2. $||\cdot |{|}_F$ is a matrix norm (frobenius norm)

3. $||\cdot |{|}_{\infty }$ is NOT a matrix norm

counterexample for (3) $A=B=$ matrix of all 1s. Then $AB$ is the matrix of all 2s.

Consider

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

References

See Also

Mentions

Mentions

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