<s>
In	O
mathematics	O
,	O
a	O
block	B-Algorithm
matrix	I-Algorithm
or	O
a	O
partitioned	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
that	O
is	O
interpreted	O
as	O
having	O
been	O
broken	O
into	O
sections	O
called	O
blocks	O
or	O
submatrices	O
.	O
</s>
<s>
Intuitively	O
,	O
a	O
matrix	B-Architecture
interpreted	O
as	O
a	O
block	B-Algorithm
matrix	I-Algorithm
can	O
be	O
visualized	O
as	O
the	O
original	O
matrix	B-Architecture
with	O
a	O
collection	O
of	O
horizontal	O
and	O
vertical	O
lines	O
,	O
which	O
break	O
it	O
up	O
,	O
or	O
partition	O
it	O
,	O
into	O
a	O
collection	O
of	O
smaller	O
matrices	O
.	O
</s>
<s>
Any	O
matrix	B-Architecture
may	O
be	O
interpreted	O
as	O
a	O
block	B-Algorithm
matrix	I-Algorithm
in	O
one	O
or	O
more	O
ways	O
,	O
with	O
each	O
interpretation	O
defined	O
by	O
how	O
its	O
rows	O
and	O
columns	O
are	O
partitioned	O
.	O
</s>
<s>
This	O
notion	O
can	O
be	O
made	O
more	O
precise	O
for	O
an	O
by	O
matrix	B-Architecture
by	O
partitioning	O
into	O
a	O
collection	O
,	O
and	O
then	O
partitioning	O
into	O
a	O
collection	O
.	O
</s>
<s>
The	O
original	O
matrix	B-Architecture
is	O
then	O
considered	O
as	O
the	O
"	O
total	O
"	O
of	O
these	O
groups	O
,	O
in	O
the	O
sense	O
that	O
the	O
entry	O
of	O
the	O
original	O
matrix	B-Architecture
corresponds	O
in	O
a	O
1-to-1	B-Algorithm
way	O
with	O
some	O
offset	B-General_Concept
entry	O
of	O
some	O
,	O
where	O
and	O
.	O
</s>
<s>
Block	B-Algorithm
matrix	I-Algorithm
algebra	O
arises	O
in	O
general	O
from	O
biproducts	O
in	O
categories	O
of	O
matrices	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
use	O
a	O
block	O
partitioned	B-Algorithm
matrix	I-Algorithm
product	O
that	O
involves	O
only	O
algebra	O
on	O
submatrices	O
of	O
the	O
factors	O
.	O
</s>
<s>
The	O
partitioning	O
of	O
the	O
factors	O
is	O
not	O
arbitrary	O
,	O
however	O
,	O
and	O
requires	O
"	O
conformable	B-Algorithm
partitions	O
"	O
between	O
two	O
matrices	O
and	O
such	O
that	O
all	O
submatrix	O
products	O
that	O
will	O
be	O
used	O
are	O
defined	O
.	O
</s>
<s>
can	O
be	O
performed	O
blockwise	O
,	O
yielding	O
as	O
an	O
matrix	B-Architecture
with	O
row	O
partitions	O
and	O
column	O
partitions	O
.	O
</s>
<s>
The	O
matrices	O
in	O
the	O
resulting	O
matrix	B-Architecture
are	O
calculated	O
by	O
multiplying	O
:	O
</s>
<s>
If	O
a	O
matrix	B-Architecture
is	O
partitioned	O
into	O
four	O
blocks	O
,	O
it	O
can	O
be	O
inverted	O
blockwise	O
as	O
follows	O
:	O
</s>
<s>
where	O
A	O
and	O
D	O
are	O
square	O
blocks	O
of	O
arbitrary	O
size	O
,	O
and	O
B	O
and	O
C	O
are	O
conformable	B-Algorithm
with	O
them	O
for	O
partitioning	O
.	O
</s>
<s>
The	O
formula	O
for	O
the	O
determinant	O
of	O
a	O
-matrix	O
above	O
continues	O
to	O
hold	O
,	O
under	O
appropriate	O
further	O
assumptions	O
,	O
for	O
a	O
matrix	B-Architecture
composed	O
of	O
four	O
submatrices	O
.	O
</s>
<s>
If	O
is	O
a	O
-matrix	O
,	O
this	O
simplifies	O
to	O
.	O
</s>
<s>
If	O
the	O
blocks	O
are	O
square	B-Algorithm
matrices	I-Algorithm
of	O
the	O
same	O
size	O
further	O
formulas	O
hold	O
.	O
</s>
<s>
A	O
block	B-Algorithm
diagonal	I-Algorithm
matrix	B-Architecture
is	O
a	O
block	B-Algorithm
matrix	I-Algorithm
that	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
such	O
that	O
the	O
main-diagonal	O
blocks	O
are	O
square	B-Algorithm
matrices	I-Algorithm
and	O
all	O
off-diagonal	O
blocks	O
are	O
zero	O
matrices	O
.	O
</s>
<s>
where	O
Ak	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
for	O
all	O
k	O
=	O
1	O
,	O
...	O
,	O
n	O
.	O
In	O
other	O
words	O
,	O
matrix	B-Architecture
A	O
is	O
the	O
direct	O
sum	O
of	O
A1	O
,	O
...	O
,	O
An	O
.	O
</s>
<s>
It	O
can	O
also	O
be	O
indicated	O
as	O
A1⊕A2⊕	O
...	O
⊕An	O
or	O
diag( A1	O
,	O
A2	O
,	O
...	O
,	O
An	O
)	O
(	O
the	O
latter	O
being	O
the	O
same	O
formalism	O
used	O
for	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
)	O
.	O
</s>
<s>
Any	O
square	B-Algorithm
matrix	I-Algorithm
can	O
trivially	O
be	O
considered	O
a	O
block	B-Algorithm
diagonal	I-Algorithm
matrix	B-Architecture
with	O
only	O
one	O
block	O
.	O
</s>
<s>
A	O
block	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
another	O
special	O
block	B-Algorithm
matrix	I-Algorithm
,	O
which	O
is	O
just	O
like	O
the	O
block	B-Algorithm
diagonal	I-Algorithm
matrix	B-Architecture
a	O
square	B-Algorithm
matrix	I-Algorithm
,	O
having	O
square	B-Algorithm
matrices	I-Algorithm
(	O
blocks	O
)	O
in	O
the	O
lower	O
diagonal	O
,	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
upper	O
diagonal	O
,	O
with	O
all	O
other	O
blocks	O
being	O
zero	O
matrices	O
.	O
</s>
<s>
It	O
is	O
essentially	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
but	O
has	O
submatrices	O
in	O
places	O
of	O
scalars	O
.	O
</s>
<s>
Block	O
tridiagonal	B-Algorithm
matrices	O
are	O
often	O
encountered	O
in	O
numerical	O
solutions	O
of	O
engineering	O
problems	O
(	O
e.g.	O
,	O
computational	O
fluid	O
dynamics	O
)	O
.	O
</s>
<s>
Optimized	O
numerical	O
methods	O
for	O
LU	O
factorization	O
are	O
available	O
and	O
hence	O
efficient	O
solution	O
algorithms	O
for	O
equation	O
systems	O
with	O
a	O
block	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
as	O
coefficient	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
Thomas	B-Language
algorithm	I-Language
,	O
used	O
for	O
efficient	O
solution	O
of	O
equation	O
systems	O
involving	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
can	O
also	O
be	O
applied	O
using	O
matrix	B-Architecture
operations	O
to	O
block	O
tridiagonal	B-Algorithm
matrices	O
(	O
see	O
also	O
Block	O
LU	O
decomposition	O
)	O
.	O
</s>
<s>
A	O
block	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
is	O
another	O
special	O
block	B-Algorithm
matrix	I-Algorithm
,	O
which	O
contains	O
blocks	O
that	O
are	O
repeated	O
down	O
the	O
diagonals	O
of	O
the	O
matrix	B-Architecture
,	O
as	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
has	O
elements	O
repeated	O
down	O
the	O
diagonal	O
.	O
</s>
<s>
A	O
special	O
form	O
of	O
matrix	B-Architecture
transpose	O
can	O
also	O
be	O
defined	O
for	O
block	B-Algorithm
matrices	I-Algorithm
,	O
where	O
individual	O
blocks	O
are	O
reordered	O
but	O
not	O
transposed	O
.	O
</s>
<s>
Let	O
be	O
a	O
block	B-Algorithm
matrix	I-Algorithm
with	O
blocks	O
,	O
the	O
block	O
transpose	O
of	O
is	O
the	O
block	B-Algorithm
matrix	I-Algorithm
with	O
blocks	O
.	O
</s>
<s>
As	O
with	O
the	O
conventional	O
trace	O
operator	O
,	O
the	O
block	O
transpose	O
is	O
a	O
linear	B-Architecture
mapping	I-Architecture
such	O
that	O
.	O
</s>
<s>
In	O
linear	B-Language
algebra	I-Language
terms	O
,	O
the	O
use	O
of	O
a	O
block	B-Algorithm
matrix	I-Algorithm
corresponds	O
to	O
having	O
a	O
linear	B-Architecture
mapping	I-Architecture
thought	O
of	O
in	O
terms	O
of	O
corresponding	O
'	O
bunches	O
 '	O
of	O
basis	O
vectors	O
.	O
</s>
<s>
That	O
again	O
matches	O
the	O
idea	O
of	O
having	O
distinguished	O
direct	O
sum	O
decompositions	O
of	O
the	O
domain	B-Algorithm
and	O
range	B-Algorithm
.	O
</s>
<s>
It	O
is	O
always	O
particularly	O
significant	O
if	O
a	O
block	O
is	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
;	O
that	O
carries	O
the	O
information	O
that	O
a	O
summand	O
maps	O
into	O
a	O
sub-sum	O
.	O
</s>
<s>
Given	O
the	O
interpretation	O
via	O
linear	B-Architecture
mappings	I-Architecture
and	O
direct	O
sums	O
,	O
there	O
is	O
a	O
special	O
type	O
of	O
block	B-Algorithm
matrix	I-Algorithm
that	O
occurs	O
for	O
square	B-Algorithm
matrices	I-Algorithm
(	O
the	O
case	O
m	O
=	O
n	O
)	O
.	O
</s>
<s>
This	O
technique	O
is	O
used	O
to	O
cut	O
down	O
calculations	O
of	O
matrices	O
,	O
column-row	O
expansions	O
,	O
and	O
many	O
computer	B-General_Concept
science	I-General_Concept
applications	O
,	O
including	O
VLSI	O
chip	O
design	O
.	O
</s>
<s>
An	O
example	O
is	O
the	O
Strassen	B-Algorithm
algorithm	I-Algorithm
for	O
fast	O
matrix	B-Architecture
multiplication	O
,	O
as	O
well	O
as	O
the	O
Hamming(7,4 )	O
encoding	O
for	O
error	O
detection	O
and	O
recovery	O
in	O
data	O
transmissions	O
.	O
</s>
<s>
For	O
example	O
,	O
The	O
matrix	B-Architecture
A	O
can	O
be	O
over	O
the	O
field	O
of	O
complex	O
numbers	O
,	O
while	O
the	O
matrix	B-Architecture
D	O
can	O
be	O
over	O
the	O
field	O
of	O
real	O
numbers	O
.	O
</s>
