Vtyjkyuk

I want to find examples of square matrices $A$ (and if possible, a general form) which satisfy the following property:

$$AA^T = frac14 left[beginmatrix
15 & 9 & 5 & -3 \
9 & 15 & 3 & -5 \
5 & 3 & 15 & -9 \
-3 & -5 & -9 & 15
endmatrixright]$$

What would a systematic way to go about this?

P.S: The matrix on the right hand side is Hermitian.

Since $B=AA^T$ is symmetric, by eigenvalues and eigenvectors, we can find $Q$ orthogonal and $Lambda$ diagonal such that

$$B=QLambda Q^T$$

and if $B$ is positive definite we have

$$B=QLambda Q^T=(QLambda^1/2)(QLambda^1/2)^T=AA^T$$

You would want this version:

$$ Q^T D Q = H $$
$$left(
beginarrayrrrr
1 & 0 & 0 & 0 \
frac 3 5 & 1 & 0 & 0 \
frac 1 3 & 0 & 1 & 0 \
- frac 1 5 & - frac 1 3 & - frac 3 5 & 1 \
endarray
right)
left(
beginarrayrrrr
15 & 0 & 0 & 0 \
0 & frac 48 5 & 0 & 0 \
0 & 0 & frac 40 3 & 0 \
0 & 0 & 0 & frac 128 15 \
endarray
right)
left(
beginarrayrrrr
1 & frac 3 5 & frac 1 3 & - frac 1 5 \
0 & 1 & 0 & - frac 1 3 \
0 & 0 & 1 & - frac 3 5 \
0 & 0 & 0 & 1 \
endarray
right)
= left(
beginarrayrrrr
15 & 9 & 5 & - 3 \
9 & 15 & 3 & - 5 \
5 & 3 & 15 & - 9 \
- 3 & - 5 & - 9 & 15 \
endarray
right)
$$

Since $D$ is positive definite, create a matrix $F$ with entries $sqrt d$ to get $Q^T F^T F Q = (FQ)^T (FQ) = H$ after deleting the $D.$ Let's see, you had a factor of $1/4,$ so $ (FQ/2)^T (FQ/2) = H/4$

Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia

$$ H = left(
beginarrayrrrr
15 & 9 & 5 & - 3 \
9 & 15 & 3 & - 5 \
5 & 3 & 15 & - 9 \
- 3 & - 5 & - 9 & 15 \
endarray
right)
$$
$$ D_0 = H $$
$$ E_j^T D_j-1 E_j = D_j $$
$$ P_j-1 E_j = P_j $$
$$ E_j^-1 Q_j-1 = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$

$$ H = left(
beginarrayrrrr
15 & 9 & 5 & - 3 \
9 & 15 & 3 & - 5 \
5 & 3 & 15 & - 9 \
- 3 & - 5 & - 9 & 15 \
endarray
right)
$$

==============================================

$$ E_1 = left(
beginarrayrrrr
1 & - frac 3 5 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
$$
$$ P_1 = left(
beginarrayrrrr
1 & - frac 3 5 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; Q_1 = left(
beginarrayrrrr
1 & frac 3 5 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; D_1 = left(
beginarrayrrrr
15 & 0 & 5 & - 3 \
0 & frac 48 5 & 0 & - frac 16 5 \
5 & 0 & 15 & - 9 \
- 3 & - frac 16 5 & - 9 & 15 \
endarray
right)
$$

==============================================

$$ E_2 = left(
beginarrayrrrr
1 & 0 & - frac 1 3 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
$$
$$ P_2 = left(
beginarrayrrrr
1 & - frac 3 5 & - frac 1 3 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; Q_2 = left(
beginarrayrrrr
1 & frac 3 5 & frac 1 3 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; D_2 = left(
beginarrayrrrr
15 & 0 & 0 & - 3 \
0 & frac 48 5 & 0 & - frac 16 5 \
0 & 0 & frac 40 3 & - 8 \
- 3 & - frac 16 5 & - 8 & 15 \
endarray
right)
$$

==============================================

$$ E_3 = left(
beginarrayrrrr
1 & 0 & 0 & frac 1 5 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
$$
$$ P_3 = left(
beginarrayrrrr
1 & - frac 3 5 & - frac 1 3 & frac 1 5 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; Q_3 = left(
beginarrayrrrr
1 & frac 3 5 & frac 1 3 & - frac 1 5 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; D_3 = left(
beginarrayrrrr
15 & 0 & 0 & 0 \
0 & frac 48 5 & 0 & - frac 16 5 \
0 & 0 & frac 40 3 & - 8 \
0 & - frac 16 5 & - 8 & frac 72 5 \
endarray
right)
$$

==============================================

$$ E_4 = left(
beginarrayrrrr
1 & 0 & 0 & 0 \
0 & 1 & 0 & frac 1 3 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
$$
$$ P_4 = left(
beginarrayrrrr
1 & - frac 3 5 & - frac 1 3 & 0 \
0 & 1 & 0 & frac 1 3 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; Q_4 = left(
beginarrayrrrr
1 & frac 3 5 & frac 1 3 & - frac 1 5 \
0 & 1 & 0 & - frac 1 3 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; D_4 = left(
beginarrayrrrr
15 & 0 & 0 & 0 \
0 & frac 48 5 & 0 & 0 \
0 & 0 & frac 40 3 & - 8 \
0 & 0 & - 8 & frac 40 3 \
endarray
right)
$$

==============================================

$$ E_5 = left(
beginarrayrrrr
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & frac 3 5 \
0 & 0 & 0 & 1 \
endarray
right)
$$
$$ P_5 = left(
beginarrayrrrr
1 & - frac 3 5 & - frac 1 3 & - frac 1 5 \
0 & 1 & 0 & frac 1 3 \
0 & 0 & 1 & frac 3 5 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; Q_5 = left(
beginarrayrrrr
1 & frac 3 5 & frac 1 3 & - frac 1 5 \
0 & 1 & 0 & - frac 1 3 \
0 & 0 & 1 & - frac 3 5 \
0 & 0 & 0 & 1 \
endarray
right)
, ; ; ; D_5 = left(
beginarrayrrrr
15 & 0 & 0 & 0 \
0 & frac 48 5 & 0 & 0 \
0 & 0 & frac 40 3 & 0 \
0 & 0 & 0 & frac 128 15 \
endarray
right)
$$

==============================================

$$ P^T H P = D $$
$$left(
beginarrayrrrr
1 & 0 & 0 & 0 \
- frac 3 5 & 1 & 0 & 0 \
- frac 1 3 & 0 & 1 & 0 \
- frac 1 5 & frac 1 3 & frac 3 5 & 1 \
endarray
right)
left(
beginarrayrrrr
15 & 9 & 5 & - 3 \
9 & 15 & 3 & - 5 \
5 & 3 & 15 & - 9 \
- 3 & - 5 & - 9 & 15 \
endarray
right)
left(
beginarrayrrrr
1 & - frac 3 5 & - frac 1 3 & - frac 1 5 \
0 & 1 & 0 & frac 1 3 \
0 & 0 & 1 & frac 3 5 \
0 & 0 & 0 & 1 \
endarray
right)
= left(
beginarrayrrrr
15 & 0 & 0 & 0 \
0 & frac 48 5 & 0 & 0 \
0 & 0 & frac 40 3 & 0 \
0 & 0 & 0 & frac 128 15 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrrr
1 & 0 & 0 & 0 \
frac 3 5 & 1 & 0 & 0 \
frac 1 3 & 0 & 1 & 0 \
- frac 1 5 & - frac 1 3 & - frac 3 5 & 1 \
endarray
right)
left(
beginarrayrrrr
15 & 0 & 0 & 0 \
0 & frac 48 5 & 0 & 0 \
0 & 0 & frac 40 3 & 0 \
0 & 0 & 0 & frac 128 15 \
endarray
right)
left(
beginarrayrrrr
1 & frac 3 5 & frac 1 3 & - frac 1 5 \
0 & 1 & 0 & - frac 1 3 \
0 & 0 & 1 & - frac 3 5 \
0 & 0 & 0 & 1 \
endarray
right)
= left(
beginarrayrrrr
15 & 9 & 5 & - 3 \
9 & 15 & 3 & - 5 \
5 & 3 & 15 & - 9 \
- 3 & - 5 & - 9 & 15 \
endarray
right)
$$

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)
$$
left(
beginarraycccc
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & -1
endarray
right)
$$

The result is a matrix in 2 by 2 blocks,
$$
M =
left(
beginarraycc
3A & A \
A & 3A
endarray
right)
$$
where
$$
A =
left(
beginarraycc
5 & 3 \
3 & 5
endarray
right)
$$
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,
$$
left(
beginarrayc
v \
v
endarray
right)
$$
has eigenvalue $8$ while
$$
left(
beginarrayc
v \
-v
endarray
right)
$$
has eigenvalue $4.$
If $w$ has eigenvalue $8,$ then
as eigenvectors for my $M$ above,
$$
left(
beginarrayc
w \
w
endarray
right)
$$
has eigenvalue $32$ while
$$
left(
beginarrayc
w \
-w
endarray
right)
$$
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 $1,2,4,8.$