How to find examples of matrices $A$ such that $AA^T$ equals a given matrix? I want to find examples of square matrices $A$ (and if possible, a general form) which satisfy the following property:
$$AA^{T} = \frac{1}{4} \left[\begin{matrix}
    15 & 9 & 5 & -3 \\
    9 & 15 & 3 & -5 \\
    5 & 3 & 15 & -9 \\
    -3 & -5 & -9 & 15
  \end{matrix}\right]$$
What would a systematic way to go about this? 
P.S: The matrix on the right hand side is Hermitian.
 A: Since $B=AA^T$ is symmetric, by eigenvalues and eigenvectors, we can find $Q$ orthogonal and $\Lambda$ diagonal such that
$$B=Q\Lambda Q^T$$
and if $B$ is positive definite we have
$$B=Q\Lambda Q^T=(Q\Lambda^{1/2})(Q\Lambda^{1/2})^T=AA^T$$
A: Looking again, this seems to be a contest type question. With a little trickery, one may find the eigenvalues entirely by hand, without attempting any 4 by 4 determinant.  MORE TO COME
First, multiply by $4,$ the fraction can be dealt with later. Next, multiply one left and right by the orthogonal matrix (its own transpose)
$$
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 
\end{array}
\right)
$$
The result is a matrix in 2 by 2 blocks,
$$
M =
\left(
\begin{array}{cc}
3A & A \\
A & 3A
\end{array}
\right)
$$
where
$$
A =
\left(
\begin{array}{cc}
5 & 3 \\
3 & 5
\end{array}
\right)
$$
The eigenvalues of this are $2,8.$ We can construct eigenvectors for the 4 by 4 $M$ above with no trouble. If $v$ has eigenvalue $2,$ then
as eigenvectors for my $M$ above,
$$
\left(
\begin{array}{c}
v \\
v
\end{array}
\right)
$$
has eigenvalue $8$ while
$$
\left(
\begin{array}{c}
v \\
-v
\end{array}
\right)
$$
has eigenvalue $4.$ 
If $w$ has eigenvalue $8,$ then
as eigenvectors for my $M$ above,
$$
\left(
\begin{array}{c}
w \\
w
\end{array}
\right)
$$
has eigenvalue $32$ while
$$
\left(
\begin{array}{c}
w \\
-w
\end{array}
\right)
$$
has eigenvalue $16.$ 
So, my $M$ has eigenvalues $4,8,16,32.$  One may use the $M$ eigenvectors to reconstruct eigenvectors for the original matrix, or start over. Including the $1/4$ fraction, the matrix in the question has eigenvalues $1,2,4,8.$
