| Version 8 |
Version 7 |
|
Suppose $M$ is a positive definite Hermitian matrix. Then $M$ has a diagonalization
|
Suppose $M$ is a positive definite Hermitean matrix. Then $M$ has a diagonalization
|
| $$ |
$$ |
| M= P^* \operatorname{diag}(\lambda_1, \ldots, \lambda_n) P |
M= P^* \operatorname{diag}(\lambda_1, \ldots, \lambda_n) P |
| $$ |
$$ |
| where $P$ is a unitary matrix and |
where $P$ is a unitary matrix and |
| $\lambda_1, \ldots, \lambda_n$ are the eigenvalues of $M$. |
$\lambda_1, \ldots, \lambda_n$ are the eigenvalues of $M$. |
|
|
| We can now define the \emph{squar{e} roo{t}} of $M$ as the matrix |
We can now define the \emph{squar{e} roo{t}} of $M$ as the matrix |
| $$ |
$$ |
| M^{1/2}= P^T \operatorname{diag}(\sqrt{\lambda_1}, \ldots, \sqrt{\lambda_n}) P. |
M^{1/2}= P^T \operatorname{diag}(\sqrt{\lambda_1}, \ldots, \sqrt{\lambda_n}) P. |
| $$ |
$$ |
| The following properties are clear |
The following properties are clear |
| \begin{enumerate} |
\begin{enumerate} |
| \item $M^{1/2} M^{1/2}=M$, |
\item $M^{1/2} M^{1/2}=M$, |
|
\item $M^{1/2}$ is Hermitian and positive definite.
|
\item $M^{1/2}$ is Hermitean and positive definite.
|
| \item $M^{1/2}$ and $M$ commute |
\item $M^{1/2}$ and $M$ commute |
| \item $(M^{1/2})^T=(M^T)^{1/2}$. |
\item $(M^{1/2})^T=(M^T)^{1/2}$. |
| \item $(M^{1/2})^{-1}=(M^{-1})^{1/2}$, so one can write $M^{-1/2}$ |
\item $(M^{1/2})^{-1}=(M^{-1})^{1/2}$, so one can write $M^{-1/2}$ |
| \item If the eigenvalues of $M$ are $(\lambda_1, \ldots, \lambda_n)$, then |
\item If the eigenvalues of $M$ are $(\lambda_1, \ldots, \lambda_n)$, then |
| the eigenvalues of $M^{1/2}$ are |
the eigenvalues of $M^{1/2}$ are |
| $(\sqrt{\lambda_1}, \ldots, \sqrt{\lambda_n})$. |
$(\sqrt{\lambda_1}, \ldots, \sqrt{\lambda_n})$. |
| \end{enumerate} |
\end{enumerate} |
