Change of Basis and Similarity — Cheat sheet

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Core rule

x=P[x]B,[x]B=P−1x,AB=P−1AP.x=P[x]_B,\quad [x]_B=P^{-1}x,\quad A_B=P^{-1}AP.

Watch for

Read matrix products from right to left. A basis change changes coordinates, not the underlying vector or operator.