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Prove that if K and L are positive definite matrices so is K + L.

My attempt:

Since K and L are positive definite we have that $q(x) > 0$ and $f(x) >0$ so $q(x) = x^TKx = \sum^n_{i,j=1} k_{ij} x_ix_j$ and $f(x) = y^TLy = \sum^n_{i,j=1}l_{ij} y_iy_j$ so if we take K + L we get that $q(x)+ f(x) = \sum^n_{i,j=1} (k_{ij} x_ix_j + l_{ij}y_iy_j)$ which is also a positive definite. Is this correct or am I way off? Can anyone show me a more simple approach for this proof?

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1 Answer 1

up vote 3 down vote accepted

$$\mathbf{x}^\mathrm{T}\left(K + L\right)\mathbf{x} =\mathbf{x}^\mathrm{T}K\mathbf{x} + \mathbf{x}^\mathrm{T}L\mathbf{x}$$ Your method is essentially the above fact elaborately written out. It is indeed correct, but perhaps a bit unnecessarily complicated.

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Thank you very much! –  diimension Oct 12 '12 at 2:53
2  
One thing I should comment on. You don't need to (more accurately, you shouldn't) use $x$ for the first function and $y$ for the second. When you combine the two, the variables need to match. –  EuYu Oct 12 '12 at 2:56
    
Thank you for pointing that out! My mistake. –  diimension Oct 12 '12 at 2:59

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