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**How can I find a solution valid for "all" cases?**

I have an equation:

[itex]tr\left\{\textbf{AB}\right\} = \sigma[/itex]

where [itex]tr\left\{\right\}[/itex] denotes the matrix trace. The square matrix [itex]\textbf{A}[/itex] is independent of both the square matrix [itex]\textbf{B}[/itex] and the real scalar [itex]\sigma[/itex].

I want to determine all possible values of [itex]\textbf{B}[/itex] that will allow the above equation to hold for

**[itex]\textbf{A}[/itex], given the only constraint:**

__all__[itex]tr\left\{\textbf{A}\right\} = 1[/itex]

For example, I can see that [itex]\textbf{B}=\sigma \textbf{I}[/itex] will always be valid (where [itex]\textbf{I}[/itex] is the identity matrix). But can I guarantee that there are no other possible values for [itex]\textbf{B}[/itex]?

I have been pondering this problem for some time and cannot see a way of approaching it. Any advice would be greatly appreciated!