Chapter 2:Generalized Inverse & Conditional Inverse
Matrix
Inverse square matrix
->
↓
↳ determinant
nonsingular matrix
must nonzero
-. An=b will have unique solution
0 x A "b
=
> CaSeS:
square matrix
3
is not
i) A involve theory
ii) A
is square but singular generalized' and 'conditional'
inverse of matrices.
Generalized Inverse (g-inverse)
Definition:
If a matrix A" exist that satisfied the four condition below,
shall call As
we a
generalized inverse of A.
i) AAY is symmetric
ii) A Ai s symmetric
iii) AAYA A =
iv) A*AAY:As
If A is nonsingular, then A "satisfies the four conditions above
Theorem: Theorem 6.5.8:
let A
be an man matrixof ranks;the
1. If a
g-inverse of an man matrix, the order nxm
2. If the null matrix Amen, then g-inverse order Anxm g-inverse of A
can be computed by following
3.
Each matrix
h as g-inverse steps:
A*A
4. Each matrix
has
uniquely-inverse
1
compute B =
5.
(AYY (A9) =
2. Let C1 = 1, I has same size as B.
6. The rank of g-inverse of Aequal to the rank of A. 3. Compute (i+i=1 ("/i) trace (ciB)-ciB
for
Rank defined to be number of
A nonzero row inthe row reduced matrik ofA . i 1.2,
=
...,r-1 (up to Cr):r-170
7. (A 9)9:A 4.
Lastly, rCrAY:Ag
8. If A
i s non singular (invertible), then A*:A tr((rB)
trace -> sum of diagonal
matrix
A
[I
:
trac A 1
=
5 3
+ +
9
=
, Example of Generalized inverse:
18)
Find the of A
-
g-inverse
Asez Axs
solutions:
Rank (A) number of
=
nonzero row in the ref(A)
107 retmem/orxex/oi)...rank (A1-2 cr 2)
=
Al
rCr AT thus, Ag=222
tr((rB) trace (CB)
B ATA =
%!i7(02):(8i]
(0,0)(same Bl
2, =
= = size as
-C,B, calculate C, B and trace cc,B)
since, C2
1(t)
= trace (CB)
(%,0);
2, B B
= =
trace,B):3
(i) trace
(!·I -1,] 3)-(,] (i)
(,B)-cB
so c= =
3 = =
1
(c)[i]:(8 1
O
=
a
Acal
Therefore:Ay =
(i)(i)
=
4
=Is) ()s =
Matrix
Inverse square matrix
->
↓
↳ determinant
nonsingular matrix
must nonzero
-. An=b will have unique solution
0 x A "b
=
> CaSeS:
square matrix
3
is not
i) A involve theory
ii) A
is square but singular generalized' and 'conditional'
inverse of matrices.
Generalized Inverse (g-inverse)
Definition:
If a matrix A" exist that satisfied the four condition below,
shall call As
we a
generalized inverse of A.
i) AAY is symmetric
ii) A Ai s symmetric
iii) AAYA A =
iv) A*AAY:As
If A is nonsingular, then A "satisfies the four conditions above
Theorem: Theorem 6.5.8:
let A
be an man matrixof ranks;the
1. If a
g-inverse of an man matrix, the order nxm
2. If the null matrix Amen, then g-inverse order Anxm g-inverse of A
can be computed by following
3.
Each matrix
h as g-inverse steps:
A*A
4. Each matrix
has
uniquely-inverse
1
compute B =
5.
(AYY (A9) =
2. Let C1 = 1, I has same size as B.
6. The rank of g-inverse of Aequal to the rank of A. 3. Compute (i+i=1 ("/i) trace (ciB)-ciB
for
Rank defined to be number of
A nonzero row inthe row reduced matrik ofA . i 1.2,
=
...,r-1 (up to Cr):r-170
7. (A 9)9:A 4.
Lastly, rCrAY:Ag
8. If A
i s non singular (invertible), then A*:A tr((rB)
trace -> sum of diagonal
matrix
A
[I
:
trac A 1
=
5 3
+ +
9
=
, Example of Generalized inverse:
18)
Find the of A
-
g-inverse
Asez Axs
solutions:
Rank (A) number of
=
nonzero row in the ref(A)
107 retmem/orxex/oi)...rank (A1-2 cr 2)
=
Al
rCr AT thus, Ag=222
tr((rB) trace (CB)
B ATA =
%!i7(02):(8i]
(0,0)(same Bl
2, =
= = size as
-C,B, calculate C, B and trace cc,B)
since, C2
1(t)
= trace (CB)
(%,0);
2, B B
= =
trace,B):3
(i) trace
(!·I -1,] 3)-(,] (i)
(,B)-cB
so c= =
3 = =
1
(c)[i]:(8 1
O
=
a
Acal
Therefore:Ay =
(i)(i)
=
4
=Is) ()s =
