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I have block matrices of the form $$\begin{pmatrix} A & v \\ v^T & x \end{pmatrix}$$ with scalar $x$ that need to be tested for positive semidefiniteness or positive definiteness.

According to J.M.'s answer here the recommended way to do this is by Cholesky decomposition, assuming nothing else is known about the matrix. However in my situation I am guaranteed that $A$ is positive semidefinite resp. positive definite.

If $A$ is positive definite then I only need to test whether $x \ge v^T A^{-1} v.$ (Is this the fastest test?)

When $A$ is positive semidefinite, it is necessary to specify that $v \in \mathrm{im}(A)$ (and this is sufficient for $v$). I am not sure what a necessary and sufficient criterion for $x$ is. Hopefully it can be done without computing a Cholesky decomposition for $A$. Thanks for any help.

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  • $\begingroup$ I forgot to mention that the entries of my matrix are integers. I don't know if that makes a difference. $\endgroup$ – user404188 Jan 4 '17 at 19:37
  • $\begingroup$ Is A sparse or dense? $\endgroup$ – Brian Borchers Jan 4 '17 at 19:39
  • $\begingroup$ @BrianBorchers $A$ is not sparse $\endgroup$ – user404188 Jan 4 '17 at 19:40
  • $\begingroup$ The quickest way to compute $v^{T}A^{-1}v$ is most likely going to be by way of the Cholesky factorization of $A$. $\endgroup$ – Brian Borchers Jan 4 '17 at 19:42

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