1. 資訊科學數學12 : Dot Product & Matrix Operation
陳光琦助理教授 (Kuang-Chi Chen)

2. Linear Equations and Matrices Dot Product and Matrix Multiplication

3. Dot Product and Matrix Multiplication
The dot product or inner product of the n-vectors a and b

4. Example 1 E.g. 1 – the dot product of u and v
u·v = (1)(2) + (-2(3) + (3)(-2) + (4)(1) = -6

5. Example 2 Example 2 Let and . If a·b = -4 , find x
a·b = 4x = -4
4x + 8 = -4
x = -3

6. Example 3 E.g. 3 – Computing a course average
w·g = (.10)(78) + (.30)(84) + (.30)(62) + (.30)(85)
= 77.1

7. Matrix Multiplication
The product of A and B, denoted by AB = C,
where
thus

8. (cont’d)
That is,
colj(B)
A B = AB
mp pn mn

9. Example 4
Example 4

10. Example 5
E.g. 5 – Compute the (3, 2) entry of AB

11. Example 6 – Linear System in Matrix Form
E.g. 6 – Linear system in matrix form
- a linear system
- a matrix form

12. Example 7 – Find x and y
E.g. 7 – Find x and y

13. AB and BA AB and BA - BA may not be defined, while BA is defined
- AB and BA are of different sizes (A23 , B32)
- AB and BA are both of the same size and they may be equal
- AB and BA are both of the same size and they may not be equal (see example)

14. Example 10 – AB ≠ BA
E.g AB ≠ BA

15. Example 11 – the 2nd column of AB
E.g the second column of AB

16. Remark Remark – inner product and matrix multiplication
If u and v are n-vectors
- Both are n1, then u·v = uTv
- Both are 1n, then u·v = uvT
- u is 1n and v is n1, then u·v = uv .

17. Example 12 – Inner Product & Matrix Multiplication
E.g. 12 – inner product and matrix multiplication

18. Matrix-vector Product
The matrix-vector product written in terms of columns

19. - A linear combination of the columns of A
(cont’d)
- A linear combination of the columns of A

20. Example 13
Example 13

21. Example 14
E.g. 14

22. Linear Systems
Linear Systems
Define

23. Linear Systems (cont’d)
We have
i.e., Ax = b
Here, A is the coefficient matrix (design matrix).

24. Augmented Matrix
Augmented Matrix

25. Example 15 – Augmented Matrix
E.g. 15

26. Example 16 – Augmented Matrix

27. Linear Equations and Matrices Properties of Matrix Operations

28. Properties of Matrix Operations
Theorem 1.1 Properties of matrix addition
A + B = B + A – commutative law
A + (B + C) = (A + B) + C – associative law
A + O = A , (O: zero matrix) – identity law
A + D = O , A + (-A) = O – additive inverse

29. Example 1 – Zero Matrix E.g. 1 - The 22 zero matrix

30. Example 2
Example 2
A + (-A) = O

31. Example 3 – Subtraction
E.g. 3 – Subtraction

32. Properties of Matrix Multiplication
Theorem 1.2 – Properties of matrix multiplication
(AB)C = A(BC) – associative law
A(B + C) = AB + AC – distributive law
(A + B)C = AC + BC .
* No commutative law !

33. Example 4 – (AB)C = A(BC) The # of multiplications to compute
A(BC) is 343 + 233 = = 54;
(AB)C is 234 + 243 = = 48.

34. Example 5
E.g. 5 – A(B + C) = AB + AC

35. Identity Matrix of Order n
Definition – Identity matrix of order n
thus, In A = A or A In = A .

36. Example 6
E.g. 6 – Identity matrix
I2A = A AI3 = A

37. Powers of A Matrix Powers of an nn matrix Definition :
thus, A0 = In ,
ApAq = Ap+q ,
(Ap)q = Apq ,
(AB)p ≠ ApBp . (why? No commu. law)

38. Example 7 – No Cancellation Law
E.g. 7 – The cancellation law doesn’t hold.
We know ab = ac and a ≠ 0 ⇒ b = c ; but it doesn’t hold in matrix. (see example)

39. Example 8
Example 8
, but B≠ C .

40. Example 9 – Markov Chain E.g. 9 – Example of Markov chain
Initial distribution
of the market

41. determine a and b so that the market is stable i.e., Ax0 = x0 .
(cont’d)
x2 = Ax1 = A(Ax0) = A2x0 .
Let x0 = [a, b]T ,
determine a and b so that the market is stable
i.e., Ax0 = x0 .

42. We have x0 = [a, b]T , a + b = 1, and Ax0 = x0 , i.e., and a = 1 – b ,
(cont’d)
We have x0 = [a, b]T , a + b = 1, and Ax0 = x0 ,
i.e., and a = 1 – b ,

43. Properties of Scalar Multiplication
Theorem Properties of scalar multiplication
r(sA) = (rs)A (…like-associative law)
r(A + B) = rA + rB (…like-distributive law)
(r + s)A = rA + sA
A(rB) = r(AB) = (rA)B .

44. Example 10
Example 10
r = -2 ,

45. Properties of Transpose
Theorem 1.4 – Properties of transpose
(AT)T = A (double transpose law)
(A + B)T = AT + BT
(AB)T = BTAT (Check? If A32 , B23)
(rA)T = rAT .

46. Example 11 – Transpose
E.g. 11 – Transpose

47. Symmetry Definition – Symmetric Let A = [aij]nn ,
If AT = A , (i.e., aij = aji ), then A is symmetric.

48. Example 12 – Symmetry
E.g. 12 – Symmetry

49. Linear Equations and Matrices Matrix Transformation

50. Matrix Transformation
R2 : denotes the set of all 2-vectors
R3 : denotes the set of all 3-vectors
Represented geometrically as directed line segments in a rectangular coordinate system

51. Vector Space – 2- & 3-vectors
(x, y)
x-axis
y-axis
Let
z-axis

52. Example 1 – n-Vectors
Example 1

53. Definition of Mapping Function
If A is an mn matrix and u is an n-vector, then Au is an m-vector ;
- Mapping Rn into Rm ;
A mapping function f : Rn → Rm defined by f(u) = Au ;
- f(u) : the image of u ;
- the range of f: the set of all images of the vectors in Rm ;
- f: the mapping function

54. u
Au = v f
f(u) = Au
f(u) = v
image
range Rn Rm

55. Example 2 – Image
Example 2

56. More Examples of Image

57. Example 3
Example 3

58. Solution of Example 3

59. Example 4 – Reflection E.g. 4 – Reflection
Let f : R2 → R2 be the matrix transformation defined by
The reflection w.r.t. the axis in R2 .

60. Example 5 – Projection Example 5 – Projection (3-dim onto 2-dim)
Let f : R3 → R2 be the matrix transformation defined by
then

61. More – Projection (3-dim onto xy-plane)
… cont’d …
More – Projection (3-dim onto xy-plane)
Let f : R3 → R3 be the matrix transformation defined by
then

62. Example 6
Example 6
Let f : R3 → R3 be the matrix transformation defined by
where r is a real number,
so we have f(u) = ru .
If r > 1, then dilation; stretches a vector;
if 0 < r < 1, then contraction; shrinks a vector.
dilation 1.膨脹，擴大。2.【醫學】擴張(症)。
stretches 1.展開，鋪開，擴張；張，繃，拉直；拉長，拉扯。2.使(精神，肌肉等)緊張，傾注全力；睜大(兩眼等)。3.勉強解釋，曲解；充分利用；亂用，濫用(法律等)；誇大(地講)。1.(時間)繼續，拖長，延長到；伸手(腳)，伸懶腰；能伸長擴張。The forest stretches for miles. 森林綿延數英里。stretch a carpet 鋪開地毯。 stretch an umbrella 撐開洋傘。
contraction 1.縮短，收縮；【醫學】攣縮。2.(開支等)縮減；收斂，狹窄；縮度。3.【語法】縮略〔如將 never 略成 ne'er, do not 略成 don't 等〕；略體，縮寫〔如 department 縮為 dep't〕。
shrink 1.皺縮；縮短，收縮。2.變小，減小。3.退縮，畏縮，害怕。1.使皺縮，弄皺；使縮短。2.縮進，拉回。

63. Example 7
Example 7
- Rotate every point in R2 counterclockwise through an angle  about the origin of a rectangular coordinate system
u = (x, y) 
f(u) = (x’, y’) 
長度 r
u = (x, y) 
f(u) = (x’, y’) 
長度 r

64. since x’ = r cos  cos  – r sin  sin  ,
(cont’d)
Let x = r cos  ,
y = r sin  ,
x’ = r cos ( + ) ,
y’ = r sin ( + ) ,
since x’ = r cos  cos  – r sin  sin  ,
and y’ = r sin  cos  + r cos  sin  .

65. so x’ = x cos  – y sin  , and y’ = x sin  + y cos  . Thus,
(cont’d)
so x’ = x cos  – y sin  ,
and y’ = x sin  + y cos  .
Thus,
where u = (x, y)T.

66. Similarly, x = x’ cos  + y’ sin  , and y = -x’ sin  + y’ cos  .
(cont’d)
Similarly, x = x’ cos  + y’ sin  ,
and y = -x’ sin  + y’ cos  .
Thus,
where u’ = (x’, y’)T.
Moreover, f’ is the inverse function of f .

67. Note: u = (x, y)  f(u) = (x’, y’)  f’(u’) = (x, y)  u’ = (x’, y’) 
(cont’d)
Note:
u = (x, y) 
f(u) = (x’, y’) 
f’(u’) = (x, y) 
u’ = (x’, y’) 