Linear algebra nugget of the day. You can classify a 2d quadratic form without finding eigenvalues. Det tells you if the eigenvalues are the same sign and if det is positive, trace tells you which sign. Exercise: what if det is 0?
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Replying to @RobJLow
Guessing without checking: if the det is zero at least one eigenvalue is zero; the other eigenvalue equals the trace.
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Replying to @johncarlosbaez @RobJLow
Strictly speaking, quadratic forms have neither determinants nor eigenvalues; only linear transformations can have determinants and eigenvalues. Quadratic forms have discriminants, which are not scalar quantities, and quadratic forms have roots.
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Replying to @MathPrinceps @RobJLow
Laurens is right. I think Robert Low secretly meant "linear transformation". That's how I read his post - I didn't really notice the phrase "quadratic form".
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Replying to @johncarlosbaez @MathPrinceps
I was assuming am inner product space with all the associated identifications. I'm a bad person.
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Replying to @RobJLow @johncarlosbaez
There's nothing wrong with assuming anything you like, but it's mildly bad not to mention what you're assuming.
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Replying to @MathPrinceps @johncarlosbaez
Can't disagree with that. Been spending to much time taking an inner product for granted recently. I'm a mildly bad person :-)
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Replying to @RobJLow @johncarlosbaez
Well, prolonged experience working in general relativity theory tends to do this to a man. I sympathize; the metric in GR is so profound and omnipresent a structure that one tends to forget what it's like not to have one.
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Replying to @MathPrinceps @RobJLow
No excuse: in GR you don't have "one" metric, you have many: one for each solution. So you gotta pay attention to which maps depend on the metric and which don't. I know you folks know this... but the kids reading this need to be told, early and often!
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Replying to @johncarlosbaez @RobJLow
I completely agree that it is wise to remain at least dimly aware of how the many geometrical structures present in each solution of the Einstein equation depend upon the spacetime metric (just as it is wise to understand Euclidean geometry as a thing of many interrelated parts.)
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This is a point first championed by Hermann Weyl, and subsequently elaborated and deepened by Ehlers-Pirani-Schild and Trautman (and, ultimately, N M J Woodhouse.) I have long wished that it were far more widely and fully understood and appreciated.
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