<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
branch	O
of	O
mathematics	O
,	O
a	O
(	O
multiplicative	O
)	O
compound	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
whose	O
entries	O
are	O
all	O
minors	O
,	O
of	O
a	O
given	O
size	O
,	O
of	O
another	O
matrix	B-Architecture
.	O
</s>
<s>
Let	O
be	O
an	O
matrix	B-Architecture
with	O
real	O
or	O
complex	O
entries	O
.	O
</s>
<s>
The	O
r	O
th	O
compound	B-Algorithm
matrix	I-Algorithm
of	O
is	O
a	O
matrix	B-Architecture
,	O
denoted	O
,	O
is	O
defined	O
as	O
follows	O
.	O
</s>
<s>
If	O
,	O
then	O
is	O
the	O
unique	O
matrix	B-Architecture
.	O
</s>
<s>
Let	O
be	O
a	O
scalar	O
,	O
be	O
an	O
matrix	B-Architecture
,	O
and	O
be	O
an	O
matrix	B-Architecture
.	O
</s>
<s>
For	O
a	O
positive	O
integer	O
,	O
let	O
denote	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
transpose	O
of	O
a	O
matrix	B-Architecture
will	O
be	O
written	O
,	O
and	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
by	O
.	O
</s>
<s>
,	O
a	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Assume	O
in	O
addition	O
that	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
size	O
.	O
</s>
<s>
Upper	B-Algorithm
triangular	I-Algorithm
,	O
</s>
<s>
Lower	B-Algorithm
triangular	I-Algorithm
,	O
</s>
<s>
Diagonal	B-Algorithm
,	O
</s>
<s>
Orthogonal	B-Algorithm
,	O
</s>
<s>
Unitary	B-Algorithm
,	O
</s>
<s>
Symmetric	B-Algorithm
,	O
</s>
<s>
Hermitian	B-Algorithm
,	O
</s>
<s>
Skew-symmetric	B-Algorithm
,	O
</s>
<s>
Skew-hermitian	O
,	O
</s>
<s>
Positive	B-Algorithm
definite	I-Algorithm
,	O
</s>
<s>
Positive	B-Algorithm
semi-definite	I-Algorithm
,	O
</s>
<s>
Normal	B-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
is	O
an	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
matrix	B-Architecture
corresponding	O
to	O
this	O
linear	B-Architecture
transformation	I-Architecture
(	O
with	O
respect	O
to	O
the	O
above	O
bases	O
of	O
the	O
exterior	O
powers	O
)	O
is	O
.	O
</s>
<s>
Let	O
be	O
an	O
matrix	B-Architecture
.	O
</s>
<s>
A	O
concrete	O
consequence	O
of	O
this	O
is	O
Jacobi	O
's	O
formula	O
for	O
the	O
minors	O
of	O
an	O
inverse	O
matrix	B-Architecture
:	O
</s>
<s>
Let	O
denote	O
the	O
sign	O
matrix	B-Architecture
:	O
</s>
<s>
and	O
let	O
denote	O
the	O
exchange	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
Then	O
Jacobi	O
's	O
theorem	O
states	O
that	O
the	O
th	O
higher	O
adjugate	O
matrix	B-Architecture
is	O
:	O
</s>
<s>
Nonetheless	O
,	O
there	O
are	O
some	O
efficient	O
algorithms	O
available	O
for	O
real	B-Architecture
matrices	I-Architecture
with	O
special	O
structure	O
.	O
</s>
