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In some presentation the speaker said that pushing AA'A-A to 0 makes encourages singular values of A to be either 0 or 1, can anyone tell me where this follows from?

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Please specify, what is $A$ exactly? (A linear transformation between inner product spaces?) Perhaps it will help to note that $(A'A)^2 = A'A$. –  Jonas Meyer Dec 9 '12 at 23:36
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up vote 6 down vote accepted

Your statement implies that $(AA')^2 = AA'$, so $AA'$ is a projection. Projections only have eigenvalues of 1 or 0. Since the singular values of $A$ are the eigenvalues of $AA'$ the original statement follows.

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