Game Development Reference
In-Depth Information
(b) Let E be equal to the identity matrix after row r has been multiplied by a sca-
lar a . Then the entries of E are given by
δ
,
if
ir
;
ij
E
=
(3.43)
ij
, f
ir
=
.
ij
The entries of the product EM are then given by
M
,
if
ir
;
n
ij
(
)
EM
=
EM
=
(3.44)
ij
ik
kj
aM
, f
i
=
r
.
ij
k
=
1
Thus, row r of the matrix M has been multiplied by a .
(c) Let E be equal to the identity matrix after row r has been multiplied by a sca-
lar a and added to row s . Then the entries of E are given by
δ
,
if
is
;
ij
E
=
(3.45)
ij
δ aδ
+
, f
is
=
.
ij
rj
The entries of the product EM are then given by
,
if
is
;
n
M
ij
(
)
EM
=
EM
=
(3.46)
ij
ik
kj
M
+
aM
, f
i
=
s
.
k
=
1
ij
rj
Thus, row r of the matrix M has been multiplied by a and added to row s .
The matrix E that represents the result of an elementary row operation per-
formed on the identity matrix is called an elementary matrix. If we have to apply
k elementary row operations to transform a matrix M into the identity matrix,
then
=
E ,
I
EE
(3.47)
kk
1
1
where the matrices
E E are the elementary matrices corresponding to
the same k row operations applied to the identity matrix. This actually shows that
the product
,
,
,
12
k
E E is equal to the inverse of M , and it is exactly what we
get when we apply the k row operations to the identity matrix concatenated to the
matrix M in Equation (3.34).
kk
1
1
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