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## Homework Statement

Not for homework, but just for understanding.

So we know that if a matrix (M) is orthogonal, then its transpose is its inverse.

Using that knowledge for a diagonalised matrix (with eigenvalues), its column vectors are all mutually orthogonal and thus you would assume that its inverse is its transpose...

However, that is wrong and the inverse is actually just the diagonalised matrix with its non-zero entries reciprocated. I understand the proof, but fail to see why the above logic wouldn't apply (beyond that it wouldn't multiply to the identity matrix)

Can anyone help me see where I have gone wrong in trying to combine the two theorems?