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Question 6.4.22: (A paradox for instructors) If AA^T = A^T A then A and A^T s...

(A paradox for instructors) If A{A}^{T} = {A}^{T} A then A and {A}^{T} share the same eigenvectors (true). A and {A}^{T} always share the same eigenvalues. Find the flaw in this conclusion: They must have the same S and \wedge. Therefore A equals {A}^{T}.

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A and {A}^{T} have the same \lambda ’s but the order of the x’s can change.

A =\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} has {\lambda}_{1} = i and {\lambda}_{2} = -i with {x}_{1} = (1, i) first for A but {x}_{1} = (1, -i) first for {A}^{T}.

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