Thus, if numeric results are acceptable, you can simply apply N to your array, and the results will automatically be orthonormal. In contrast, with floating-point results, such performance considerations do not apply, and so by default it orthonormalizes them. As a result, Mathematica does not normalize symbolic eigenvectors because doing so could be catastrophic.
Why is Mathematica not smart enough to do this? For exact input, the resulting eigenvectors can be complex numeric expressions, and calculating norms and other such arithmetic used in the Gram-Schmidt procedure becomes frightening from a computation-time perspective. While the documentation does not specifically say that symbolic Hermitian matrices are not necessarily given orthonormal eigenbases, it does sayįor approximate numerical matrices m, the eigenvectors are normalized.įor exact or symbolic matrices m, the eigenvectors are not normalized.įrom this, it's reasonable to guess that if Mathematica isn't smart enough to normalize symbolic results, then it's also probably not smart enough to orthonormalize them, either.
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