<s>
A	O
square	B-Algorithm
root	I-Algorithm
of	I-Algorithm
a	I-Algorithm
2×2	I-Algorithm
matrix	I-Algorithm
M	O
is	O
another	O
2×2	O
matrix	B-Architecture
R	O
such	O
that	O
M	O
=	O
R2	O
,	O
where	O
R2	O
stands	O
for	O
the	O
matrix	B-Architecture
product	O
of	O
R	O
with	O
itself	O
.	O
</s>
<s>
In	O
many	O
cases	O
,	O
such	O
a	O
matrix	B-Architecture
R	O
can	O
be	O
obtained	O
by	O
an	O
explicit	O
formula	O
.	O
</s>
<s>
Square	O
roots	O
that	O
are	O
not	O
the	O
all-zeros	O
matrix	B-Architecture
come	O
in	O
pairs	O
:	O
if	O
R	O
is	O
a	O
square	O
root	O
of	O
M	O
,	O
then	O
−R	O
is	O
also	O
a	O
square	O
root	O
of	O
M	O
,	O
since	O
( −R	O
)	O
( −R	O
)	O
=	O
( −1	O
)	O
( −1	O
)	O
(	O
RR	O
)	O
=	O
R2	O
=	O
M.A	O
2×2	O
matrix	B-Architecture
with	O
two	O
distinct	O
nonzero	O
eigenvalues	O
has	O
four	O
square	O
roots	O
.	O
</s>
<s>
A	O
positive-definite	B-Algorithm
matrix	I-Algorithm
has	O
precisely	O
one	O
positive-definite	O
square	O
root	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
general	O
formula	O
that	O
applies	O
to	O
almost	O
any	O
2	O
×	O
2	O
matrix	B-Architecture
.	O
</s>
<s>
Note	O
that	O
R	O
may	O
have	O
complex	O
entries	O
even	O
if	O
M	O
is	O
a	O
real	O
matrix	B-Architecture
;	O
this	O
will	O
be	O
the	O
case	O
,	O
in	O
particular	O
,	O
if	O
the	O
determinant	O
δ	O
is	O
negative	O
.	O
</s>
<s>
The	O
formula	O
above	O
fails	O
completely	O
if	O
δ	O
and	O
τ	O
are	O
both	O
zero	O
;	O
that	O
is	O
,	O
if	O
D	O
=	O
−A	O
,	O
and	O
A2	O
=	O
−BC	O
,	O
so	O
that	O
both	O
the	O
trace	O
and	O
the	O
determinant	O
of	O
the	O
matrix	B-Architecture
are	O
zero	O
.	O
</s>
<s>
If	O
M	O
is	O
an	O
idempotent	O
matrix	B-Architecture
,	O
meaning	O
that	O
MM	O
=	O
M	O
,	O
then	O
if	O
it	O
is	O
not	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
its	O
determinant	O
is	O
zero	O
,	O
and	O
its	O
trace	O
equals	O
its	O
rank	O
,	O
which	O
(	O
excluding	O
the	O
zero	O
matrix	B-Architecture
)	O
is	O
1	O
.	O
</s>
<s>
If	O
the	O
matrix	B-Architecture
M	O
can	O
be	O
expressed	O
as	O
real	O
multiple	O
of	O
the	O
exponent	O
of	O
some	O
matrix	B-Architecture
A	O
,	O
,	O
then	O
two	O
of	O
its	O
square	O
roots	O
are	O
.	O
</s>
