Subsection
{\sc\large This Section is a Draft, Subject to Changes}\\
{\sc\large Needs Numerical Examples}
\bigskip
With all our results about Hermitian matrices, their eigenvalues and their diagonalizations, it will be a nearly trivial matter to now construct a “square root” of a positive semi-definite matrix. We will describe the square root of a matrix $A$ as a matrix $S$ such that $A=S^2$. In general, a matrix $A$ might have many such square roots. But with a few results in hand we will be able to impose an extra condition on $S$ that will make a unique $S$ such that $A=S^2$. At that point we can define
the square root of $A$ formally.
\subsect{SRM}{Square Root of a Matrix}
Theorem PSMSR Positive Semi-Definite Matrices and Square Roots
Suppose $A$ is a square matrix. There is a positive semi-definite matrix $S$ such that $A=S^2$ if and only if $A$ is positive semi-definite.
Proof
There is a very close relationship between the eigenvalues and eigenspaces of a positive semi-definite matrix and its positive semi-definite square root. The next theorem is interesting in its own right, but is also an important technical step in some other important results, such as the upcoming uniqueness of the square root (Theorem
USR).
Theorem EESR Eigenvalues and Eigenspaces of a Square Root
Suppose that $A$ is a positive semi-definite matrix and $S$ is a positive semi-definite matrix such that $A=S^2$. If $\scalarlist{\lambda}{p}$ are the distinct eigenvalues of $A$, then the distinct eigenvalues of $S$ are $\sqrt{\lambda_1},\,\sqrt{\lambda_2},\,\sqrt{\lambda_3},\,\dots,\,\sqrt{\lambda_p}$, and $\eigenspace{S}{\sqrt{\lambda_i}}=\eigenspace{A}{\lambda_i}$ for $1\leq i\leq p$.
Proof
Notice that we defined the singular values of a matrix $A$ as the square roots of the eigenvalues of $\adjoint{A}A$ (Definition
SV). With Theorem
EESR in hand we recognize the singular values of $A$ as simply the eigenvalues of $\sr{\adjoint{A}A}$. Indeed, many authors take this as the definition of singular values, since it is equivalent to our definition. We have chosen not to wait for a discussion of square roots before making a definition of singular values, allowing us to present the singular value decomposition (Theorem
SVD) all the sooner.
In the first half of the proof of Theorem
PSMSR we could have chosen the matrix $E$ (which was the essential component of the desired matrix $S$) in a variety of ways. Any collection of diagonal entries of $E$ could be replaced by their negatives and we would maintain the property that $E^2=D$. However, if we decide to enforce the entries of $E$ as non-negative quantities then $E$ is positive semi-definite, and then $S$ follows along as a positive semi-definite matrix. We now show that of all the possible square roots of a positive semi-definite matrix, only one is itself again positive semi-definite. In other words, the $S$ of Theorem
PSMSR is unique.
Theorem USR Unique Square Root
Suppose $A$ is a positive semi-definite matrix. Then there is a unique positive semi-definite matrix $S$ such that $A=S^2$.
Proof
With a criteria that distinguishes one square root from all the rest (positive semi-definiteness) we can now define
the square root of a positive semi-definite matrix.
Definition SRM Square Root of a Matrix
Suppose $A$ is a positive semi-definite matrix and $S$ is the positive semi-definite matrix such that $S^2=SS=A$. Then $S$ is the square root of $A$ and we write $S=\sr{A}$.
$\sr{A}$
$\square$
