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Let $M$ be a positive definite matrix over complex field and let $\mu$ be its smallest eigenvalue. Is the matrix $M-\frac{\mu}{2}I$ positive definite? Thanks in advance.

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There is a diagonal matrix $\;D\;$ with all its diagonal entries the given matrix's eigenvalues, and inversible $\;P\;$ such that

$$P^{-1}MP=D\implies P^{-1}\left(M-\frac\mu2I\right)P=D-\frac\mu2I$$

and since all the entries on the diagonal of $\;D-\frac\mu2I\;$ are positive, the matrix $\;M-\frac\mu2\;$ is positive definite.

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    $\begingroup$ Is it obvious that $M$ can be diagonalised? $\endgroup$ – πr8 Dec 17 '16 at 21:06
  • $\begingroup$ @πr8 Of course it is: it is a symmetric matrix, which are the ones for which the notion of "positive definite" and etc. is defined, and any symmetric matrix is not only diagonalizable but even orthogonally diagonalizable. $\endgroup$ – DonAntonio Dec 17 '16 at 21:13
  • $\begingroup$ Right, to me it wasn't clear that it was necessarily symmetric. Agreed on / aware of all other points. $\endgroup$ – πr8 Dec 17 '16 at 21:15

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