# Can I use set notation that accounts for multiplicity?

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 $\{1,1,1,2,3,3,4,4,4,4\}$.)

• What do you want to do next? That is what for do you need this notation?
– quid
Sep 13, 2016 at 16:02

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.

• But couldn't I just use $\{\lambda_i, \mu_i \}$? Sep 13, 2016 at 15:57
• @user58865 no you cannot. As $\{2,3\} =\{3,2\}$ and you can hardly be indifferent to a having EV 2 with mult 3 or EV 3 with mult 2.
– quid
Sep 13, 2016 at 16:06
• Ah, good point @quid - thanks... Sep 13, 2016 at 16:11

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.

• I don't need ordering of the eigenvalues, actually...and regarding eigenvalues, I wonder whether set notation is more appropriate than list notation... Sep 13, 2016 at 15:55
• The ordering is because the first element of your pair is the eigenvalue and the second element is the multiplicity. If you use braces, convention is that $\{3,1\} = \{1,3\}$, but in your proposed notation they mean different things. So order matters. Sep 13, 2016 at 16:25

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?

• Right, I don't need ordering... Sep 13, 2016 at 15:56