# Positive definite matrix and it's eigenvalue.

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.

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.
• Is it obvious that $M$ can be diagonalised? – πr8 Dec 17 '16 at 21:06