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Recently, I com across a questions, which can be given as follows,

If there are two symmetric positive semi-definite matrix $W$ and $T$, but they satisfy the following condition: $null(W)\cap null(T) = \{0\}$, where $null(·)$ denotes the null space of the corresponding matrix.

Then can we obtain that one of these two matrices is symmetric positive definite?

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    $\begingroup$ Consider matrices $\left[\matrix{1 & 0\\0 & 0}\right]$ and $\left[\matrix{0 & 0\\0 & 1}\right]$. $\endgroup$ – A.Γ. Jul 13 '15 at 9:08
  • $\begingroup$ Yeah, I see. Many thanks. $\endgroup$ – Hsien-Ming Ku Jul 13 '15 at 14:20

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