introduction to matrices and vectors sebastian van delden usc upstate [email protected]
TRANSCRIPT
Introduction
mnmm
ij
n
n
aaa
a
aaa
aaa
A
21
22221
11211
m rows
n columns
mn matrix
element in ith row, jth column
When m = n, A is called a square matrix.
Also written as A=aij
Definition: A matrix is a rectangular array of numbers.
Matrix Equality Definition: Let A and B be two matrices. These
matrices are the same, that is, A = B if they have the same number of rows and columns, and every element at each position in A equals the element at corresponding position in B.
* This is not trivial if elements are real numbers subject to digital approximation.
The Transpose of a Matrix
mnnn
m
m
T
nmnn
m
m
T
xxx
xxx
xxx
xxx
xxx
xxx
21
22212
12111
21
22221
11211
X
63
52
41
654
321
TA
A
142
314
143
214
TB
B
Note that (XT)T = X
5
Matrix Addition, SubtractionLet A = aij , B = bij be mn matrices. Then:A + B = aij + bij, and A –– B = aij –– bij
34
04
34
32
41
43
02
43
11
30
82
52
32
41
43
02
43
11
Properties of Matrix Addition
Commutative: A + B = B + A
Associative: A + (B + C) = (A + B) + C
Inventories Makealot, Inc. manufactures widgets, nerfs,
smores, and flots. It supplies three different warehouses (#1,#2,#3).
Opening inventory:
20
50
25
10
55
12
33
90
245
89
6
15
025
3
35
630
10
2 412
3
550
0 7
20
6 380
041
77
3
Sales: Closing inventory:
– =
w n s f
#1
#2
#3
Scalar Multiplication
8070
6050
4030
2010
87
65
43
21
10
Associative: c1(c2A) = (c1c2)A
Distributive: (c1 + c2) A = c1A + c2A
Matrix MultiplicationLet A be an mk matrix, and B be a kn matrix. Then their product is: AB=[cij]
1 1 2 21
k
ij it tj i j i j ik kjt
c a b a b a b a b
24232221
14131211
34333231
24232221
14131211
232221
131211
cccc
cccc
bbbb
bbbb
bbbb
aaa
aaa
12321322121211 cbababa
Matrix MultiplicationLet A be an mk matrix, and B be a kn matrix. Then their product is: AB=[cij]
1 1 2 21
k
ij it tj i j i j ik kjt
c a b a b a b a b
24232221
14131211
34333231
24232221
14131211
232221
131211
cccc
cccc
bbbb
bbbb
bbbb
aaa
aaa
322212 cbababa 21 22 23 22
Matching Dimensions
24232221
14131211
34333231
24232221
14131211
232221
131211
cccc
cccc
bbbb
bbbb
bbbb
aaa
aaa
To multiply two matrices, the dimensions must match:
23 34
have to be equal
24 matrix
23 34 24
8 dot products
Multiplicative PropertiesNote even if AB is defined, BA might not be.Example: If A is 34, B is 46, then AB is a 36 matrix, but BA is not defined.Even if both AB and BA are defined, they may not havethe same dimensions. Even if they do, the result might not be equal.
23
34
35
23
11
12
12
11
BAAB
BAHowever, provided that the dimensions match, (AB)C = A(BC)
Chained Matrix Multiplication
13 5 5 89 89 3 3 34
What is the most efficient way of carrying out the
following chained matrix multiplication?
M
Chained Matrix Multiplication
1 2 3 413 5 5 89 89 3 3 34
What is the most efficient way of carrying out the
following chained matrix multiplication.
M M M M M
1 413 5 5 3 3 34
413 3 3 34
13 34
Let's try:
5 89 3 1335
13 5 3 195
13 3 34 1326
Total multiplications = 1335 195 1326 2856.
M M M
M M
M
Chained Matrix Multiplication
1 2 3 413 5 5 89 89 3 3 34
What is the most efficient way of carrying out the
following chained matrix multiplication.
M M M M M
1 2 3 413 5 5 89 89 3 3 34
Answer:
This would require 2856 multiplications
M M M M M
Example
9 5 5 2 2 6
What is the most efficient way of carrying out the
following chained matrix multiplication?
M
Example
9 5 5 2 2 6
9 5 5 2 2 6
9 5 5 2 2 6
What is the most efficient way of carrying out the
following chained matrix multiplication?
:
If we do the cost is
5 2 6 9 5 6 330
If we do the cost is
9 5
Answe
2 9 2
r
M
6 198. So, this is the optimal way.
Ways of Parenthesizing a product of n matrices Let T(n) be the number of essentially distinct ways of parenthesizing a product of n matrices. The values of T(n) are known as Catalan numbers. Here are few values of T(n):
n 1 2 3 4 5 … 10 … 15 T(n) 1 1 2 5 14 … 4862 … 2674440
It can be shown that T(n) = Ω(22n/n2)
Identity Matrix
100
010
001
3I
The identity matrix is a square matrix with all 1’s alongthe diagonal and 0’s elsewhere. Example:
fe
dc
ba
fe
dc
ba
0000
0000
0000
100
010
001
For an mn matrix A, Im A = A In
(mm) (mn) = (mn) (nn)
Inverse Matrix
Let A and B be nn matrices. If AB=BA=In then B is called the inverse of A,
denoted B=A-1.
Not all square matrices are invertible.
Symmetric Matrix
If matrix A is such that A = AT then it is called a symmetric matrix. For example:
1 4 1
4 3 0
1 0 2
is symmetric. Note, for A to be symmetric, is has to be square. Note also thatNote also that I Inn is trivially symmetric.
Vectors
m
2
1
a
a
a
a ]bbb[
q21b
An m element column vector A q element row vector
Transpose the column Transpose the row
].[ 21 mT aaaa
q
T
b
b
b
b
2
1
Vectors A 1xN or Nx1 matrix
1xN is called a row vector Nx1 is called a column vector N is the dimension of the vector
Vectors can be drawn as arrows and so have a direction and a magnitude.
Magnitude:
222
21 ... naaa
na
a
a
given
2
1
a
Drawing Vectors
x
y
a = (8,5)
8
5
Unit Vectors Magnitude is 1 A normalized vector is a unit vector that has be obtained by divided
each dimension of a vector by its magnitude. It has the same direction as the original vector. Important because something direction is all that is important – magnitude is not
needed…
x
a = (8,5)
8
5
y |a| = sqrt(82 + 52) =~ 9.4Normalized a, a’ = (8/9.4, 5/9.4) = (.85, .53)
a’ = (.85, .53)
Geometry of Vectors
x
y
(a, b)
a
b
If m is magnitude:
a = m . cos , b = m . sin
For unit vectors:
a = cos , b = sin
= tan-1(b/a)
Addition- preserves direction and magnitude.- application: robot position translations- tip to tail method:
x
y
v
u
u + v
Subtraction- application: can represent robot position error vector
- u – v, a vector originating in v and ending in u
x
y
v
uu - v
Multiplication with a scalar- can change magnitude and direction (if multiplied with a negative number.
x
y
v
u
½ v
-u
Cross Product
Produces a vector perpendicular (normal) to the plane created by the 2 vectors.
u x v
v
u
u x v
Cross product
Direction is determined by the right hand rule Put hand on first vector (left side of x) and curl
fingers towards second vector. Magnitude of u x v is |u| . |v| . sin(theta)
where theta is the angle between u and v So, cross product produces a vector
Dot product Length of the projection of one vector onto a another.
u . v Dot product is |u| . |v| . cos(theta) where theta is the
angle between u and v So, dot product produces a scalar Note: is u and v and unit vectors, the dot product is
simply: cos(theta) u
v
cos (theta)
theta
Dot and cross products Dot product from unit vectors:
As angle approaches 0, dot product approaches 1 As angle approaches 90, dot product approaches 0
Cross product from unit vectors: As angle approaches 0, dot product approaches 0 As angle approaches 90, dot product approaches 1
Finally…. Perpendicular vectors (dot product
= 0) are called orthogonal vectors. Orthogonal unit vectors are called
orthonormal vectors. Think: what do you need to
represent a 3D coordinate system…? Three orthonormal vectors: X, Y, and Z….