5
$\begingroup$

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?

https://en.wikipedia.org/wiki/Matrix_norm

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.