Proving Hermition Matrix Real & Eigenvalues/Eigenspinors

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in the spin matrix if the <A> is real how we can prove the matrix is herimition ,and how we can prove the eigenvalues of any hermition 2x2 matrix is real and its eigenspinors are orthognal if the eigenvalues is defferent and if it is the same? and A physical quantity
 
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Thread 'Is a table levitating possible, even at a minuscule probability?'

I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
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