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If the quadratic form is > 0, then it’s positive definite. If the quadratic form is ≥ 0, then it’s positive semi-definite. If the quadratic form is < 0, then it’s negative definite.
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Solution 1. I don't think there is a nice answer for matrices in general. Most often we care about positive definite matrices for Hermitian matrices, so a lot is known in this case.
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A matrix is positive definite fxTAx > Ofor all vectors x 0. Frequently in physics the energy of a system in state x is represented as XTAX (or XTAx) and so this is frequently called the energy
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