<s>
In	O
mathematics	O
,	O
a	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
is	O
an	O
matrix	B-Architecture
representing	O
the	O
spacing	O
of	O
a	O
set	O
of	O
points	O
in	O
Euclidean	O
space	O
.	O
</s>
<s>
For	O
points	O
in	O
-dimensional	O
space	O
,	O
the	O
elements	O
of	O
their	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
are	O
given	O
by	O
squares	O
of	O
distances	O
between	O
them	O
.	O
</s>
<s>
Euclidean	O
distance	O
matrices	O
are	O
closely	O
related	O
to	O
Gram	B-Algorithm
matrices	I-Algorithm
(	O
matrices	O
of	O
dot	O
products	O
,	O
describing	O
norms	O
of	O
vectors	O
and	O
angles	O
between	O
them	O
)	O
.	O
</s>
<s>
The	O
latter	O
are	O
easily	O
analyzed	O
using	O
methods	O
of	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
A	O
realization	O
,	O
if	O
it	O
exists	O
,	O
is	O
unique	O
up	O
to	O
rigid	B-Algorithm
transformations	I-Algorithm
,	O
i.e.	O
</s>
<s>
distance-preserving	O
transformations	O
of	O
Euclidean	O
space	O
(	O
rotations	B-General_Concept
,	O
reflections	B-Algorithm
,	O
translations	B-Algorithm
)	O
.	O
</s>
<s>
The	O
goal	O
may	O
be	O
to	O
visualize	O
such	O
data	O
by	O
points	O
in	O
Euclidean	O
space	O
whose	O
distance	O
matrix	B-Architecture
approximates	O
a	O
given	O
dissimilarity	O
matrix	B-Architecture
as	O
well	O
as	O
possible	O
—	O
this	O
is	O
known	O
as	O
multidimensional	O
scaling	O
.	O
</s>
<s>
Alternatively	O
,	O
given	O
two	O
sets	O
of	O
data	O
already	O
represented	O
by	O
points	O
in	O
Euclidean	O
space	O
,	O
one	O
may	O
ask	O
how	O
similar	O
they	O
are	O
in	O
shape	O
,	O
that	O
is	O
,	O
how	O
closely	O
can	O
they	O
be	O
related	O
by	O
a	O
distance-preserving	B-Algorithm
transformation	I-Algorithm
—	O
this	O
is	O
Procrustes	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Some	O
of	O
the	O
distances	O
may	O
also	O
be	O
missing	O
or	O
come	O
unlabelled	O
(	O
as	O
an	O
unordered	O
set	O
or	O
multiset	O
instead	O
of	O
a	O
matrix	B-Architecture
)	O
,	O
leading	O
to	O
more	O
complex	O
algorithmic	O
tasks	O
,	O
such	O
as	O
the	O
graph	O
realization	O
problem	O
or	O
the	O
turnpike	O
problem	O
(	O
for	O
points	O
on	O
a	O
line	O
)	O
.	O
</s>
<s>
By	O
the	O
fact	O
that	O
Euclidean	O
distance	O
is	O
a	O
metric	O
,	O
the	O
matrix	B-Architecture
has	O
the	O
following	O
properties	O
.	O
</s>
<s>
it	O
is	O
a	O
hollow	B-Algorithm
matrix	I-Algorithm
)	O
;	O
hence	O
the	O
trace	O
of	O
is	O
zero	O
.	O
</s>
<s>
is	O
symmetric	B-Algorithm
(	O
i.e.	O
</s>
<s>
In	O
dimension	O
,	O
a	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
has	O
rank	O
less	O
than	O
or	O
equal	O
to	O
.	O
</s>
<s>
Distances	O
can	O
be	O
shrunk	O
by	O
any	O
power	O
to	O
obtain	O
another	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
if	O
is	O
a	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
,	O
then	O
is	O
a	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
for	O
every	O
.	O
</s>
<s>
is	O
the	O
matrix	B-Architecture
of	O
their	O
dot	O
products	O
(	O
here	O
a	O
point	O
is	O
thought	O
of	O
as	O
a	O
vector	O
from	O
0	O
to	O
that	O
point	O
)	O
:	O
</s>
<s>
Thus	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
describes	O
norms	O
and	O
angles	O
of	O
vectors	O
(	O
from	O
0	O
to	O
)	O
.	O
</s>
<s>
Let	O
be	O
the	O
matrix	B-Architecture
containing	O
as	O
columns	O
.	O
</s>
<s>
Matrices	O
that	O
can	O
be	O
decomposed	O
as	O
,	O
that	O
is	O
,	O
Gram	B-Algorithm
matrices	I-Algorithm
of	O
some	O
sequence	O
of	O
vectors	O
(	O
columns	O
of	O
)	O
,	O
are	O
well	O
understood	O
—	O
these	O
are	O
precisely	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
contains	O
additional	O
information	O
:	O
distances	O
from	O
0	O
.	O
</s>
<s>
is	O
called	O
a	O
realization	O
of	O
in	O
if	O
is	O
their	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
One	O
can	O
assume	O
without	O
loss	O
of	O
generality	O
that	O
(	O
because	O
translating	B-Algorithm
by	O
preserves	O
distances	O
)	O
.	O
</s>
<s>
This	O
follows	O
from	O
the	O
previous	O
discussion	O
because	O
is	O
positive	B-Algorithm
semidefinite	I-Algorithm
of	O
rank	O
at	O
most	O
if	O
and	O
only	O
if	O
it	O
can	O
be	O
decomposed	O
as	O
where	O
is	O
an	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
two	O
main	O
approaches	O
are	O
variants	O
of	O
Cholesky	O
decomposition	O
or	O
using	O
spectral	O
decompositions	O
to	O
find	O
the	O
principal	O
square	O
root	O
of	O
,	O
see	O
Definite	B-Algorithm
matrix	I-Algorithm
#Decomposition	O
.	O
</s>
<s>
A	O
more	O
symmetric	B-Algorithm
variant	O
of	O
the	O
same	O
theorem	O
is	O
the	O
following	O
:	O
</s>
<s>
In	O
particular	O
,	O
these	O
allow	O
to	O
show	O
that	O
a	O
symmetric	B-Algorithm
hollow	B-Algorithm
matrix	B-Architecture
is	O
realizable	O
in	O
if	O
and	O
only	O
if	O
every	O
principal	O
submatrix	O
is	O
.	O
</s>
<s>
Two	O
sets	O
of	O
points	O
are	O
called	O
homometric	O
if	O
they	O
have	O
the	O
same	O
multiset	O
of	O
distances	O
(	O
but	O
are	O
not	O
necessarily	O
related	O
by	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
)	O
.	O
</s>
<s>
Given	O
a	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
,	O
the	O
sequence	O
of	O
points	O
that	O
realize	O
it	O
is	O
unique	O
up	O
to	O
rigid	B-Algorithm
transformations	I-Algorithm
–	O
these	O
are	O
isometries	O
of	O
Euclidean	O
space	O
:	O
rotations	B-General_Concept
,	O
reflections	B-Algorithm
,	O
translations	B-Algorithm
,	O
and	O
their	O
compositions	O
.	O
</s>
<s>
Rigid	B-Algorithm
transformations	I-Algorithm
preserve	O
distances	O
so	O
one	O
direction	O
is	O
clear	O
.	O
</s>
<s>
Without	O
loss	O
of	O
generality	O
we	O
can	O
assume	O
by	O
translating	B-Algorithm
the	O
points	O
by	O
and	O
,	O
respectively	O
.	O
</s>
<s>
Then	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
remaining	O
vectors	O
is	O
identical	O
to	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
vectors	O
(	O
)	O
.	O
</s>
<s>
This	O
implies	O
there	O
exists	O
an	O
orthogonal	B-Algorithm
matrix	B-Architecture
such	O
that	O
,	O
see	O
Definite	B-Algorithm
symmetric	I-Algorithm
matrix	I-Algorithm
#Uniqueness	O
up	O
to	O
unitary	O
transformations	O
.	O
</s>
<s>
describes	O
an	O
orthogonal	B-Algorithm
transformation	O
of	O
(	O
a	O
composition	O
of	O
rotations	B-General_Concept
and	O
reflections	B-Algorithm
,	O
without	O
translations	B-Algorithm
)	O
which	O
maps	O
to	O
(	O
and	O
0	O
to	O
0	O
)	O
.	O
</s>
<s>
The	O
final	O
rigid	B-Algorithm
transformation	I-Algorithm
is	O
described	O
by	O
.	O
</s>
<s>
In	O
applications	O
,	O
when	O
distances	O
do	O
n't	O
match	O
exactly	O
,	O
Procrustes	B-Algorithm
analysis	I-Algorithm
aims	O
to	O
relate	O
two	O
point	O
sets	O
as	O
close	O
as	O
possible	O
via	O
rigid	B-Algorithm
transformations	I-Algorithm
,	O
usually	O
using	O
singular	O
value	O
decomposition	O
.	O
</s>
<s>
The	O
ordinary	O
Euclidean	O
case	O
is	O
known	O
as	O
the	O
orthogonal	B-Algorithm
Procrustes	O
problem	O
or	O
Wahba	O
's	O
problem	O
(	O
when	O
observations	O
are	O
weighted	O
to	O
account	O
for	O
varying	O
uncertainties	O
)	O
.	O
</s>
