This document provides an overview of matrix algebra concepts including: (1) How to represent systems of linear equations using matrix notation as AX = B where A is the coefficient matrix, X is the vector of unknowns, and B is the vector of constants. (2) The definitions and properties of matrix operations such as addition, scalar multiplication, and matrix multiplication. (3) The concepts of identity matrices, null matrices, transpose of a matrix, and inverse of a matrix and their properties. (4) How to solve systems of linear equations using the inverse of the coefficient matrix such that if A is invertible, the solution is X = A^-1B.

1. shito@seinan-gu.ac.jp
2020 5 1
•
•
•
(1) 2
(2)
(3) 2
非線形曲線 (nonlinear curve)
線型近似 (linear approximation)
1

2. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
1 (Matrix and Vector)
(1)
(2)
• 2
3x + y = 4
9x − 3y = 8
=⇒
3 1
9 −3
x
y
=
4
8
3
•
a11x1 + a12x2 + · · · + a1nxn = d1
a21x1 + a22x2 + · · · + a2nxn = d2
...
...
...
...
am1x1 + am2x2 + · · · + amnxn = dm
(a) x
(b)
(c)
A = x = d =
II 2

3. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
• A
A = (aij) (i = 1, 2, · · · , m, j = 1, 2, · · · , n)
(row) (column)
• A = (aij) i j
• × aij
• m = n
(3)
• (column vector): ::::::
1
• (row vector): ::::::
1
II 3

4. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
•
1
1
n
⇓
n n– n
n– 1
• x d
• x d
•
x
2×2
=
x11 x12
x21 x22
=⇒
d
m×1
=






d1
d2
...
dm






=⇒
(4)
1 Ax = d 2
(a) 2 A x
(b) Ax d
1
⇓
(Matrix Algebra)
II 4

5. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
(1) pp.65–69
(2) 4.1 (p.69) 1 2
2
(1)
A = (aij) B = (bij)
A = B ⇐⇒ aij = bij for all i and j.
4 3
0 2
=
4 3
0 2
=
3 4
0 2
x
y
=
7
4
x = y =
(2)
•
•
3 8
9 5
2×2
+
4 1
2 7
2×2
=



a11 a12
a21 a22
a31 a32



3×2
−



b11 b12
b21 b22
b31 b32



3×2
=
(3)
=⇒
7
3 −1
0 5
=
1
2
a11 a12
a21 a22
=
II 5

6. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
(4)
• AB
a11 a12
a21 a22



b11 b12
b21 b22
b31 b32


 =
a11 a12
a21 a22
b11 b12 b13
b21 b22 b23
=
4 3
2 5
3
2
=
9 3
2 7
8 3 2
4 1 5
=
4 3 5



2
9
6


 =
(5)
(6)
: x1 + x2 + x3 =
3
j=1 xj
Quiz
(a)
5
i=2
yi =
(b) x0 + x1 + x2 + x3 =
(c)
3
j=1
ajxj =
(d)
n
i=0
aixi
=
(e)
3
k=1
axk = a
3
k=1
xk
II 6

7. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
• xi i xi
•
a11 a12
a21 a22
b11 b12
b21 b22
=
c11 c12
c21 c22
cij =
2
k=1
aikbkj (i = 1, 2, j = 1, 2)
c11 =
c12 =
c21 =
c22 =
(1) pp.70–78
(2) 4.2 1–7
II 7

8. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
3
(1)
u =



4
2
5


 v =



2
1
2



u v =
uv =
(2)
2 n n
(a) u =
3
2
or u = ( 3 2 )
u u (0, 0) u or u 2
(3, 2) 1 u
II 8

9. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
(b)
(c) v =
1
4
u =
3
2
u + v =
v − u =
II 9

10. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
(d) 1 1
3v + 2u = 3
1
4
+ 2
3
2
=
9
16
1 2
n
i=1
kivi = k1v1 + k2v2 + · · · + knvn
ki vi
(3) 1 (linearly dependent) 1 (linearly independent)
1 v1, · · · , vn 1 1
1
1
3 1 v1 =
2
7
v2 =
1
8
v3 =
4
5
1 1
3v1 − 2v2 = v3
↔ 3v1 − 2v2 − v3 = 0 0 ( 0 0 )
1 1
2 m v1, · · · , vn
n
i=0 kivi = 0m×1
1 k1, · · · , kn
1 i ki = 0
1
II 10

11. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
(4) 1
v1 =
3
2
v2 =
6
4
v1 =
3
2
v2 =
−3
−2
2
=⇒
(5) 1
v1 =
1
4
, v2 =
3
2
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12. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
v1 v2 1
⇓
2 2 1
1
⇓
2 3 1
1
(6)
(a) 2 u v 1 2
• 1
• 2
• 2 1 u v 2 2
1
• 1 2
1
• 2
1 2
2
2 2
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13. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
• ( 1 0 ) ( 0 1 ) i 1 0
(b) 3 3 3
3 p.87 4.4
(c) n- n- n-
n- 1 n n
(1) pp.79–88
(2) 4.3 1–6 (p.89)
4
a b A B
: a + b = b + a
: ab = ba
: (a + b) + c = a + (b + c)
: (ab)c = a(bc)
: a(b + c) = ab + ac
(1) pp.90–93
(2) 4.4 1–5
II 13

14. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
5
(1) (Identity Matrices)
1
I
2×2
= I
3×3
=
I
4×4
=
• 1
1 × a = a × 1 = a
IA = AI = A
A
2×3
=
1 2 3
2 0 3
I
2×2
A
2×3
=
A
2×3
I
3×3
=
• A
m×n
I
n×n
B
n×p
=
• (I)2
=
II 14

15. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
(2) (Null Matrices)
O =
0 0 0
0 0 0
• A
m×n
+ O
m×n
= O
m×n
+ A
m×n
= A
m×n
• A
m×n
O
n×p
= O
m×p
O
q×m
A
m×n
= O
q×n
(3)
• ab = 0 a b
AB =
2 4
1 2
−2 4
1 −2
=
• cd = ce d = e c = 0
C =
2 3
6 9
, D =
1 1
1 2
, E =
−2 1
3 2
CD = CE =
(1) pp.94–96
(2) 4.5 1–3
6
(1) (transposed matrices)
A =
2 5
4 9
A = AT
=
B
m×n
B n×m
II 15

16. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
(2)
(A ) =
(A + B) =
(AB) =
(3) (inverse matrices)
• A
• AA−1
= A−1
A =
• A A
⇐⇒ A
⇐⇒ A
•
(A−1
)−1
= A
• A n × n A−1
n × n
•
• (AB)−1
=
• (A )−1
=
II 16

17. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00
(4)
A
(n×n)
x
(n×1)
= d
(n×1)
(1) pp.97–103
(2) 4.6 1–5
II 17