How can we transform matrices into scalars?

Question

If we have the three matrices:

$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} , \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{bmatrix} , \text{ and } \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{bmatrix} $$

How can I transform the matrices so that the first one (the identity) becomes a constant (scalar) $n$, and the other two become 0?

I'm not exactly sure what I'm looking for, but I'm looking for as many ways as possible to do this transformation with the hope that I'll find a way that works. Again, I'm looking for a way to transform the matrices into a scalar.

Answer 1

The most obvious one to me seems like the trace operation, i.e. the sum of the elements along the leading diagonal. For example: $$\text{tr}\left(\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\right) = 1+1+1 = 3$$ $$\text{tr}\left(\left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right]\right) = 0+0+0 = 0$$ $$\text{tr}\left(\left[\begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]\right) = 0+0+0 = 0$$ The trace is important for many reasons:

• The trace is independent of changes of bases.
• The trace is an additive homomorphism: $\text{tr}(X+Y) = \text{tr}(X)+\text{tr}(Y)$.