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January 16th, 2017, 07:07 AM   #1
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From: london

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Matrix - positive definite

u is unit vector, i.e$\displaystyle ||u|| =1$, $\displaystyle u^Tu=1$
c is a positive constant
M is a matrix
I is the identity matrix

$\displaystyle M = I + cuu^T$

In the first part of the question I had to calculate the matrix square root $\displaystyle M^{1/2}$ which has form

$\displaystyle M^{1/2} = pI + quu^T$

From the answers I know that

$\displaystyle p = \pm 1$
$\displaystyle 2pq + q^2 = c $

How can I prove that $\displaystyle M^{1/2}$ is *not* positive definte symmetric when p = -1?

How can I prove that $\displaystyle M^{1/2}$ is positive definite symmetric when p = 1?
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