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9
Linear algebra
Read: Boas Ch. 3.
9.1 Properties of and operations with matrices
M
×
N
matrix with elements
A
ij
A
=
A
11
A
12
... A
1
j
... A
1
N
A
21
A
22
... A
2
j
... A
2
N
.
.
.
.
.
.
.
.
.
.
.
.
A
i
1
A
i
2
... A
ij
... A
iN
.
.
.
.
.
.
.
.
.
.
.
.
A
M
1
A
M
2
... A
Mj
... A
MN
(1)
Deﬁnitions:
•
Matrices are equal if their elements are equal,
A
=
B
⇔
A
ij
=
B
ij
.
•
(
A
+
B
)
ij
=
A
ij
+
B
ij
•
(
kA
)
ij
=
kA
ij
for
k
const.
•
(
AB
)
ij
=
∑
N
‘
=1
A
i‘
B
‘j
.
Note for multiplication of rectangular matrices, need
(
M
×
N
)
·
(
N
×
P
).
•
Matrices need not “commute”.
AB
not nec. equal to
BA
. [
A,B
]
≡
AB

BA
is called “commutator of
A
and
B
. If [
A,B
] = 0,
A
and
B
commute.
•
For square mats.
N
×
N
, det
A
=

A

=
∑
π
sgn
π A
1
π
(1)
A
2
π
(2)
...A
Nπ
(
N
)
,
where
sum is taken over all permutiations
π
of the elements
{
1
,...N
}
. Each term
in the sum is a product of
N
elements, each taken from a diﬀerent row of
A
and from a diﬀerent column of
A
, and sgn
π
. Examples:
ﬂ
ﬂ
ﬂ
ﬂ
A
11
A
12
A
21
A
22
ﬂ
ﬂ
ﬂ
ﬂ
=
A
11
A
22

A
12
A
21
,
(2)
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
=
A
11
A
22
A
33
+
A
12
A
23
A
31
+
A
13
A
21
A
32

A
12
A
21
A
33

A
13
A
22
A
31

A
11
A
22
A
32
(3)
•
det
AB
= det
A
·
det
B
but det(
A
+
B
)
6
= det
A
+ det
B
. For practice with
determinants, see Boas.
1
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Identity matrix
I
:
IA
=
A
∀
A
.
I
ij
=
δ
ij
.
•
Inverse of a matrix.
A
·
A

1
=
A

1
A
=
I
.
•
Transpose of a matrix (
A
T
)
ij
=
A
ji
.
•
Formula for ﬁnding inverse:
A

1
=
1
det
A
C
T
,
(4)
where
C
is “cofactor matrix”. An element
C
ij
is the determinant of the
N

1
×
N

1 matrix you get when you cross out the row and column (i,j), and
multiply by (
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 Fall '07
 HIRSHFELD

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