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Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
7.1 Matrices, Vectors: Addition and Scalar Multiplication
Matrix ; rectangular array of numbers (or functions) enclosed in brackets.
(1)
• entry, element
• row, column
• square matrix
• vector ; matrix having a single row or column - row vector, column vector
16 2 0 0
5 1 3 0
. .
33 32 31
23 22 21
13 12 11
a a a
a a a
a a a
x x e
x x
4 2
2
6
2
a1 a2 a3
5 0
4 .
a 23 ; read a two three
Ex. 1) Linear Systems 6 9 6
4x1 x2 x3 20 2
6x1 x3 10 8
5x1 x2 x3
coefficient matrix,
A
and augmented matrix A~
1 8 5
2 0
6
9 6 4 A
10 1 8 5
20 2 0
6
6 9 6 4 A~
General Concepts and Notations
Matrix is denoted by capital boldface letters A, B, C, ∙·· or by general entry in bracket, thus A An m x n matrix means a matrix with m rows and n columns.
].
[
= ajk
are denoted by
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
(2)
mn m
m
n n
jk
a a
a
a a
a
a a
a a
A
2 1
2 22
21
1 12
11
Each entry in (2) has two subscripts. The first is the row number and the second is the column number. Thus, is the entry in row
2 and column 1. 21
a
If m=n , A is an n x n square matrix and is called the main diagonal of A. A matrix that is not square matrix is called a rectangular matrix.
ann
a a
a11, 22, 33,,
Vectors
A vector is a matrix with only row or column. It is denoted by lowercase boldface letters a, b, … or by a = [ajk ]
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
• row vector
bm
b b
b .
2 1
a a an
a 1 2 a
2 5 0.8 0 1
• column vector
7 0 4 - b
Matrix Addition and Scalar Multiplication Equality of Matrices
Two matrices A = and B = are equal if and only if they have the same size and the corresponding entries are equal.
]
[ajk [bjk]
Ex 3) Equality of Matrices
Let A
22 21
12 11
a a
a
a and
1 3
0 B 4
Then A = B if and only if
1 3
0 4
22 21
12 11
a a
a a
, , Ex.)
Ex.)
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Addition of Matrices
The sum of two matrices A= and B= of the same size is written A + B and has the entries ajk bjk. Matrices of different sizes cannot be added.
Ex 4) Addition of Matrices and Vectors
Scalar Multiplication (Multiplication by a Number)
The product of any matrix A = and any scalar c (number c ) is written cA= obtained by multiplying each entry of A by c.
2 1 0
3 6 A 4
If and ,
0 1 3
0 1
5
B then
3 2 2
3 5 B 1
A
If a
5 7 2
and b
6 2 0
, then a b
1 9 2
][ajk [bjk]
] [ajk ]
[cajk
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Ex 5) Scalar Multiplication
If ,
5 . 4 0 . 9
9 . 0 0
8 . 1 7 . 2
A then ,
5 . 4 0 . 9
9 . 0 0
8 . 1 7 . 2
A
5 10
1 0
2 3 9
10 A
0 0
0 0
0 0 0 A
Rules for Matrix Addition and Scalar Multiplication
(3)
(a) A + B = B + A commutative
(b) (A + B) + C = A + (B + C) associative
(written A + B + C) (c) A + 0 = A
(d) A + (-A) = 0
* 0 denotes the zero matrix (of size m×n).
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Similarly, for scalar multiplication,
(4)
(a) c(A + B) = cA + cB (b) (c+k)A = cA + kA (c) c(kA) = (ck)A (d) 1A = A
7.2 Matrix Multiplication
Multiplication of a Matrix by a Matrix
The product C = AB of an m×n matrix A = times anr×pmatrix B = is defined if and only if n=r and is then the m×p matrix C = with entries
(1)
n
l
nk jn k
i k j lk jl
jk a b a b a b a b
c
1
2 2 1
1
p k
m j
, , 2 , 1
, , 2 , 1
]
[ajk [bjk]
] [cjk
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
The condition n=r means that the second matrix, B must have as many rows as the first matrix, A has columns.
A B = C
[m x n] [r x p] = [m x p]
in (1) is obtained by multiplying each entry in the j th row of A by the corresponding entry in the k th column of B and then adding these n products.
cjk
← multiplication of rows into columns.
43 42
41
33 32
31
23 22
21
13 12
11
a a
a
a a
a
a a
a
a a
a
32 31
1 22 2
12 11
b b
b b
b b
=
4×3 3×2 4×2
42 41
32 31
22 21
12 11
c c
c c
c c
c c
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Ex. 1) Matrix Multiplication
28 37
4 9
6 14 16 26
42 43
2 22
1 1 4 9
8 7 0 5
1 3 2 2
2 3 6
2 0 4
1 5 3
Ex. 2) Multiplication of a Matrix and a Vector
43 22 5
8 3 1
5 2 3 4 5
3 8 1
2 4
Ex. 3) Products of Row and Column Vectors
4 24 12
2 12 6
1 6 3 1 6 3 4 2 1 19
4 2 1 1 6 3
Ex. 4) Matrix multiplication is not commutative, AB ≠ BA in general.
0 0
0 0 1 1
1 1 100 100
1
1 but
99 99
99 99
100 100
1 1 1 1
1 1
* AB = 0 does not necessarily imply that BA = 0 or A = 0 or B = 0
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Matrix multiplication satisfies the following rules.
(2)
(a) (kA)B = k(AB) = A(kB) written kAB or AkB
(b) A(BC) = (AB)C written ABC, associative law (c) (A + B)C = AC + BC distributive law
(d) C(A + B) = CA + CB distributive law
Matrix multiplication is a multiplication of rows into columns, (3) cjk ajbk j 1,,m; k 1,, p
where is the j th row vector of A and is the k th column vector of B.aj
b
k
j k j k jn nknk k jn j
j k
j a b a b a b
b b a
a a b
a
1 1 2 2
1 2
1
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Transposition
Transposition of Matrices and Vectors
The transpose of an m x n matrix A = is the n x m matrix AT (read A transpose) that has the first row of A as its first column, the second row of A as its second column, and so on. Thus, AT =
] [ajk
(9) AT =
mn n
n
m m
kj
a a
a
a a
a
a a
a a
2 1
2 22
12
1 21
11
Ex. 7) Transposition of Matrices and Vectors
If ,
0 0 4
1 8
5
A
0 1
0 8
4 5 AT
then A little more compactly,
3 2 6 3 2 6 1 ,
0 8 3 1 8
0 , 3
0 1
0 8
4 5 0
0 4
1 8
5 T T T
] [akj
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Rules for transposition are
(10)
(a) (AT)T = A
(b) (A+B)T= AT+BT (c) (cA)T = cAT
(d) (AB)T = BTAT * transposed matrix is in reversed order
Special Matrices
Symmetric and Skew-Symmetric Matrices
AT = A (thusakj ajk ), AT = -A (thus akj ajk , henceajj 0) (11)
Symmetric Matrix Skew-Symmetric Matrix
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Ex. 8) Symmetric and Skew-Symmetric Matrices
30 150 200
150 10 120
200 120 20
A is symmetric and
0 2 3
2 0 1
3 1 0
B is skew-symmetric
Triangular Matrices
Upper Triangular Matrices, Lower Triangular Matrices Ex. 9) Upper and Lower Triangular Matrices
6 3 9 1
0 2 0 1
0 0 3 9
0 0 0 3 , 8 6 7
0 1 8
0 0 2 6
0 0
2 3 0
2 4 1 2 ,
0 3 1
Upper triangular Lower triangular
Chap. 7 Linear Algebra : Matrices, Vectors, Determinants
Diagonal Matrices
* Diagonal Matrix D, Scalar matrix S, Unit matrix I (12) AS = SA = cA
(13) AI = IA = A
Ex. 10) Diagonal Matrix D, Scalar matrix S, Unit matrix I
1 0 0
0 1 0
0 0 1 ,
0 0
0 0
0 0 ,
0 0 0
0 3 0
0 0 2
I c c c S D
•In case all the diagonal entries of a diagonal matrix S are equal to c
