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Mixing positive definite matrices
If A and B are positive definite matrices of the same size and C=Ay+B(1-y) where y is a scaler, then is C also positive definite?
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- adannenberg
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Mixing positive definite matrices
it looks like y is supposed to be a wt, but you don't say that. the answer is obviously no if y outside of [0,1] is allowed. consider A = 2x2 identity, B = 2x2 with 0 diagonal terms, -3 off-diagonal. y = 4. then c = ((4,9),(9,4)) with det = 16-81...
Mixing positive definite matrices
It is trivial to show that for a positive definite matrix, A, and for a real scalar, a>0, aA is also positive definite. It is also easy to show that a sum of two positive definite matrices is also positive definite. Therefore, if 0<y<1 your C matrix will be positive definite.The reverse does not generally hold, I think. For instance, if A-B is positive definite then C will be positive definite for every y>0. But if you're doing what I think you're doing then you should be fine with that requirement.
Last edited by flapjack on April 14th, 2011, 10:00 pm, edited 1 time in total.