- #1

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I always get confused with these basis-change problems. First of all: What does it even mean that a matrix is represented in a particular basis? Why is H necessarily represented in the basis (1,0) and (0,1)?

Secondly I know that the basis change equation is:

H' = SHS^-1 (1)

But my QM also tells me that the elements of H' can be found by taking the inner products <a'lHla'>, <a'lHlb'> etc. How is this equivalent to (1)?