If A is a $k \times k$ positive definite matrix and b is $k \times 1$ vector, are all elements in A b always positive?
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What is a positive vector? – José Carlos Santos Jun 06 '19 at 22:07
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I meant a vector with all elements are positive. – Art1 Jun 06 '19 at 22:08
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2Consider this: if $A$ is positive definite, and has full rank, then for an arbitrary $b$, the entries of $Ab$ are also arbitrary. – Not Legato Jun 06 '19 at 22:19
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The answer is no.
Here is an example:
$$A =\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$
$$\mathbf{b} =\begin{pmatrix} \phantom{-}0 \\ -1 \end{pmatrix}$$
$$A\mathbf{b} =\begin{pmatrix} \phantom{-}0 \\ -1 \end{pmatrix}$$
mjw
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No. Take $A=\left(\begin{matrix}1&1\\-1&1\end{matrix}\right)$ and $v=\left(\begin{matrix}1\\0\end{matrix}\right)$. I will leave it to you to prove that above the reals $A$ is positive definite.
eranreches
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