# Matrix Theory: An essential proof for eigenvector computations

I’ve avoided proofs unless absolutely necessary, but the relation between the same eigenvector expressed in two different bases, is important.
Given that AS is the linear transformation matrix in standard basis S, and AB is its counterpart in basis B, we can write the relation between them as:

$A_B = C^{-1}A_SC\\* A_S = CA_BC^{-1}$

where C is the similarity transformation. We’ve seen this relation already; check here if you’ve forgotten about it.
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# Tests increase our Knowledge of a System: A Probabilistic Proof

This was an old proof that was up on my old blog, but since I’m no longer posting to that, I’m reposting it here for posterity. Also, rewriting the equations in LaTeX, now that I have installed a plugin for that.

I present a simple mathematical device to prove that tests improve our understanding of code. It does not really matter if this is code written by the test author himself or is legacy. To do this, some simplification of the situation is necessary.
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