File: One Step Inverse Updating
Consider that we have an
m
m
×
matrix
1
B
that we have computed the inverse (here we are
assuming that it exists)
1
1
B
−
.
We could have done this inverse by the product form method:
1
1
1
2
1
m
m
B
M M
M M
−
−
=
"
Now assume that we want to replace column
i
of
1
B
with the column vector
a
.
The reason for
this exchange in a simplex type algorithm is to change basic variables.
So let’s denote the two
matrices by
1
2
1
1
2
2
,
,
,
,
,
,
,
, ,
,
i
m
m
B
a
a
a
a
B
a
a
a
a
⎡
⎤
=
⎣
⎦
⎡
⎤
=
⎣
⎦
"
"
"
"
Now we also have
1
1
B
−
such that
1
2
1
2
1
1
1
1
1
1
2
1
2
1
1
1
2
1
,
,
,
,
,
,
,
,
,
,
,
,
, ,
,
,
,
,
,
,
i
m
i
m
m
m
B
B
B
a
a
a
a
e
e
e
e
B
B
B
a
a
a
a
e
e
e
α
−
−
−
−
⎡
⎤
⎡
⎤
=
=
⎣
⎦
⎣
⎦
⎡
⎤
⎡
⎤
=
=
⎣
⎦
⎣
⎦
"
"
"
"
"
"
"
"
The if we form an
M
matrix such that
1
2
1
2
,
,
,
,
,
,
,
,
,
,
m
i
m
M
e
e
e
e
e
e
e
α
⎡
⎤
⎡
⎤
=
⎣
⎦
⎣
⎦
"
"
"
"
, then
1
1
2
1
B
MB
−
−
=
since
1
2
1
1
1
2
1
1
2
1
2
,
,
, ,
,
,
,
,
,
,
,
,
,
,
,
m
m
i
m
MB
B
MB
a
a
a
a
M
e
e
e
e
e
e
e
α
−
−
⎡
⎤
=
⎣
⎦
⎡
⎤
⎡
⎤
=
=
⎣
⎦
⎣
⎦
"
"
"
"
"
"