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.
2026-04-25 10:07:09.1777111629
Positive definite matrix and it's eigenvalue.
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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.