Can I use set notation that accounts for multiplicity?

Question

Can I use $\{\lambda_i,\mu_i\}$ to describe a matrix's eigenvalues and account for their multiplicity?

Here $\lambda_i$ would be the eigenvalues, and $\mu_i$ would be its multiplicity.

(I need the multiplicity of the eigenvalues in my work, in set notation, but don't want to use the notation such as $\{1,1,1,2,3,3,4,4,4,4\}$.)

Answer 1

It's not appropriate since sets are inherently unordered. You could use an ordered pair $(\lambda_i,\mu_i)$. Since you have a list of eigenvalues with their multiplicity, a set of ordered pairs

$$\left\{(\lambda_1,\mu_1),(\lambda_2,\mu_2), \dotsc, (\lambda_n,\mu_n)\right\} $$

would probably be best.

Answer 2

You're sort of after a "multiset". (See https://en.wikipedia.org/wiki/Multiset.) The typical notation for $\{2,2,2,5,5\}$ would be $\{(2,3),(5,2)\}.$ Since $\{*\}$ is used for un-ordered pairs and $(*)$ for ordered pairs, then the parens would be more appropriate.

Answer 3

You could also consider a family, which takes account for multiplicity:

$$(2,2,2,5,5)$$

but it also takes account for order, so may be it is not what you're looking for?