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Chapter 2
Matrices and Linear Algebra
2.1 Basics
Definition 2.1.1. A matrix is an m×n array of scalars from a given field
F. The individual values in the matrix are called entries.
Examples. ^ ^
A= 213 B= 12
−124 34
The size of the array is–written as m×n,where
m×n
cA
number of rows number of columns
Notation
a a ... a
11 12 1n
A
a a ... a ←− rows
21 22 2n
A= t
a a ... a
n1 n2 mn
AAc
columns
A:=uppercase denotes a matrix
a := lower case denotes an entry of a matrix a ∈ F.
Special matrices
33
34 CHAPTER2. MATRICESANDLINEARALGEBRA
(1) If m = n, the matrix is called square.Inthiscasewehave
(1a) A matrix A is said to be diagonal if
a =0 i W= j.
ij
(1b) A diagonal matrix A may be denoted by diag(d ,d ,...,d )
1 2 n
where
a =d a =0 j W= i.
ii i ij
Thediagonalmatrixdiag(1,1,...,1)is called the identity matrix
and is usually denoted by
10... 0
01
I =
n . .
. ..
.
01
or simply I,whenn is assumed to be known. 0 = diag(0,...,0)
is called the zero matrix.
(1c) A square matrix L is said to be lower triangular if
f =0 ij.
ij
(1e) A square matrix A is called symmetric if
a =a .
ij ji
(1f) A square matrix A is called Hermitian if
a =¯a (¯z := complex conjugate of z).
ij ji
(1g) E has a 1 in the (i,j) position and zeros in all other positions.
ij
(2) A rectangular matrix A is called nonnegative if
a ≥0alli,j.
ij
It is called positive if
a >0alli,j.
ij
Eachofthesematriceshassomespecialproperties, whichwewill study
during this course.
2.1. BASICS 35
Definition 2.1.2. The set of all m × n matrices is denoted by M (F),
m,n
where F is the underlying field (usually R or C). In the case where m = n
we write M (F) to denote the matrices of size n×n.
n
Theorem 2.1.1. M is a vector space with basis given by E , 1 ≤ i ≤
m,n ij
m, 1≤j≤n.
Equality, Addition, Multiplication
Definition 2.1.3. Two matrices A and B are equal if and only if they have
thesamesizeand
a =b all i, j.
ij ij
Definition 2.1.4. If A is any matrix and α ∈ F then the scalar multipli-
cation B = αA is defined by
b =αa all i, j.
ij ij
Definition 2.1.5. If A and B are matrices of the same size then the sum
AandB is defined by C = A+B,where
c =a +b all i, j
ij ij ij
Wecan also compute the difference D = A−B by summing A and (−1)B
D=A−B=A+(−1)B.
matrix subtraction.
Matrix addition “inherits” many properties from the field F.
Theorem 2.1.2. If A,B,C ∈ M (F) and α,β ∈ F,then
m,n
(1) A+B=B+A commutivity
(2) A+(B+C)=(A+B)+C associativity
(3) α(A+B)=αA+αB distributivity of a scalar
(4) If B =0(a matrix of all zeros) then
A+B=A+0=A
(4) (α+β)A=αA+βA
36 CHAPTER2. MATRICESANDLINEARALGEBRA
(5) α(βA)=αβA
(6) 0A =0
(7) α0=0.
Definition 2.1.6. If x and y ∈ R ,
n
x=(x ...x )
1 n
y =(y ...y ).
1 n
Then the scalar or dot product of x and y is given by
n
x,yX = 3x y .
i i
i=1
Remark 2.1.1. (i) Alternate notation for the scalar product: x,yX = x·y.
(ii) The dot product is defined only for vectors of the same length.
Example 2.1.1. Let x =(1,0,3,−1) and y =(0,2,−1,2) then x,yX =
1(0) +0(2)+3(−1)−1(2) = −5.
Definition 2.1.7. If A is m×n and B is n×p.Letr (A) denote the vector
i
th
with entries given by the i row of A,andletc (B) denote the vector with
j
entries given by the jth row of B. The product C = AB is the m×p matrix
defined by
c =r(A),c(B)X
ij i j
th
where r (A) is the vector in R consisting of the i row of A and similarly
i n
c (B) is the vector formed from the jth column of B. Other notation for
j
C=AB
n
c = a b 1 ≤ i ≤ m
ij ik kj
k=1
1 ≤ j ≤ p.
Example 2.1.2. Let
} ]
101 21
30
A= 321 and B= .
−11
Then
} ]
12
AB= 11 4 .
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