When do we use matrix norm? matrix norm is one of the property of a matrix, but I am not sure when I will use it. Do we use it for calculating a upper bound of a matrix?



Matrix norms can be a way to state that "a matrix is big".

In statistics, per example. If you are trying to compare variances of multiple estimators $\hat\theta_1,\hat\theta_2$ of a multivariate $\theta$ you need to compare $var(\hat\theta_1)$ and $var(\hat\theta_2)$. As they are matrices, they may not be comparable. Looking at their norms can be a way to compare them.

In numerical analysis matrix norms can provide useful inequalities when looking for eigenvalues. Per example :

Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an operator in $\mathbb{R}^n$ given by $x \mapsto Ax$. Prove that $||A|| = \mathrm{max}_j |\lambda_j|$, where $\lambda_j$ are the eigenvalues of $A$.

With more details here:https://math.stackexchange.com/questions/603375/norm-of-a-symmetric-matrix-equals-spectral-radius

Edit, following the comments If you are able to prove $||A||<1$, then a sequence defined by $x_{n+1}=Ax_{n}$ converges to 0.

  • $\begingroup$ Thanks for the answer. So, why do we want to know the max value of norm is equal to max value of eigenvalues? $\endgroup$ – RockTheStar Oct 1 '15 at 17:48
  • $\begingroup$ The fact that $|\lambda_{max}|<1$ (or the opposite) can have a lot of implications on what you are studying... Looking at the norm is cheaper than evaluating all the eigen values. $\endgroup$ – RUser4512 Oct 2 '15 at 9:45
  • $\begingroup$ So, what are the implications? $\endgroup$ – RockTheStar Oct 4 '15 at 0:12

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