2
$\begingroup$

In a book I have been reading recently a question as follows came up as a problem and I am unsure how to solve it:

Two quantities are represented by the matrices

$ \text{M = } \left[\begin{array}{rrr} 3 & 0 & -i \\ 0 & 1 & 0\\ i & 0 & 3 \end{array}\right] $

$ \text{N = } \left[\begin{array}{rrr} 3 & 0 & 2i \\ 0 & 7 & 0\\ -2i & 0 & 3 \end{array}\right] $

The possible values of the quantity represented by M are 1, 2 and 4.

What are the possible values of the quantity represented by N?

Explain how you know that.

Any help would be greatly appreciated!

  • 0
    Without the context given by the book, "quantity represented by $M$" is really vague. In fact, even **with** the book, it's unclear what this means. And it's certainly not standard language in mathematics.2012-04-17

1 Answers 1

8

Note that the matrices $M$ and $N$ are self-adjoint. Given that you're reading a book on quantum mechanics, it makes sense to look at their spectrum (set of eigenvalues).

The spectrum of $M$ is $\{1,2,4\}$ and the spectrum of $N$ is $\{1,5,7\}$.

  • 0
    @F$i$xee I'll certainly use that method when I don't have the energy to do it myself! :)2012-04-18