Question

If the trace of AB is 25 then the trace of BA is​

Answer 1

Answer:

Let's write the trace in a more convenient way. We have:

Aei=⎡⎣⎢⎢a11⋮an1⋯⋱⋯a1n⋮ann⎤⎦⎥⎥⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢0⋮1⋮0⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥=⎡⎣⎢⎢ai1⋮ain⎤⎦⎥⎥,

where the 1 is in the i-th entry. This way:

⟨ei,Aei⟩=etiAei=aii.

So tr(A)=∑iaii. Now: (AB)ij=∑kaikbkj, and:

tr(AB)=∑i∑kaikbki.

On the other hand, (BA)ij=∑kbikakj. So:

tr(BA)=∑i∑kbikaki.

They are the same quantity, up to renaming indices (i↔k)

Step-by-step explanation:

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