Download Lecture 1 Matrix Algebra

Econ 306
Lecture 1
Matrix Algebra
Lecture Outline
1. DEFINITION
2. ADDITION AND SUBTRACTION
3. MULTIPLICATION
4. THE TRACE OF A MATRIX
5. THE DETERMINANT OF A SQUARE MATRIX
6. THE RANK OF A MATRIX
7. INVERSE MATRICIES AND GENERASLISED
INVERSES
8. IDEMPOTENT MATRICES
9. KRONECKER PRODUCT
 a11
a
 22
 .

A=  .
am1
a12 . . a1n 
a22 . . . 
. . . . 

. . . . 
. . . amn 
= A[aij]
To transpose a matrix:
 a11 a12
A
a 21 a 22
a13 
a 23 
 a11
A  AT  a12
a13
a 21 
a 22 
a 23 
If A = AT the matrix is SYMMETRIC
A special symmetric matrix is the IDENTITY MATRIX
1 0 . . 0 
0 1 . . . 


I  . . . . .


.
.
.
.
.


0 0 . . 1
1 0
I2  

0
1


The identity matrix is also an example of a DIAGONAL
MATRIX
diag (a1 , a 2 ,..., a n )
For A = B both matrices must be of the same size and
aij = bij
Equality is TRANSITIVE
C=A+B
cij=aij+bij
 6 8
A

  2 4
0  3
B

9 7 
6 5 
C

7 11
Matrices must be CONFORMABLE
Matrix addition and subtraction are COMMUTATIVE
A B  B A
and ASSOCIATIVE
( A  B)  C  A  ( B  C )
also
( A  B)  A  B
T
T
T
SCALAR multiplication
B = kA
bij = kaij
1 2
A

3 4
k=2
 2 4
B

6
8


Two forms of multiplication
C = AB
C = BA
A premultiplies B
A postmultiplies B
For multiplication to be possible the number of columns
in A must equal the number of rows in B
A = (m x p)
B = (p x n)
p
cij   aik bkj
k 1
C = AB = (m x n)
 6 8
A

  2 4
 3  8 1
D

9 2 5
F = AD
(6  8)  (8  2)
(6  1)  (8  5) 
 (6  3)  (8  9)
(2  3)  (4  9) (2  8)  (4  2) (2  1)  (4  5)


90  32 46
30 24 18 


In general
AB  BA
except when
Both matrices are diagonal
Scalar multiplication
When one matrix is an identity matrix
AN INNER PRODUCT is the product of a row vector
and a column vector
J = (T x 1) with all elements 1
JTJ = T
Two vectors are ORTHOGONAL IF THEIR INNER
PRODUCT =0
 2
a 
5 
5
b 
  2
aTb = 0
The inner product is commutative aTb = bTa
√of the inner product of a vector with itself is the
EUCLIDEAN VECTOR NORM
For multiplication the DISTRIBUTIVE law holds
A(B + C) = AB + AC
As does the associative law
A(BC) = (AB)C
The CANCELLATION law does not hold
If AB = 0, then it does not follow that A or B or both are
0
1 2
 2  4
A
B


 1 2 
3 6
The TRANSPOSE OF A PRODUCT
(AB)T=BTAT
If A = AT and B = BT then (AB)T = BTAT = BA does not
necessarily = AB
If A is symmetric but B is not note that BTAB is
symmetric since (BTAB)T = BTATB = BTAB
If A = I then BTB is symmetric
The TRACE of a matrix tr(A) = a11 + a22 + …. + ann
tr(AT) = tr(A)
tr(kA) = ktr(A)
tr(In) = n
If A and B are square
tr(A + B) = tr(A) + tr(B)
tr(AB) = tr(BA)
tr(ABC) = tr(BCA) = tr(CAB)
The DETERMINANT of a matrix det A or │A│
 a11 a12 
A

a
a
22 
 21
│A│= a11 - a12
 a11
A  a 21
a31
a12
a 22
a32
a13 
a 23 
a33 
A  a11
a22
a23
a32
a33
 a12
a21 a23
a31
a33
 a13
a21 a22
a31
a32
a11( a22a33-a23a32) - a12(a21a33-a23a31) + a13(a21a32-a22a31)
( a22a33-a23a32) is called the minor of a11 and is usually
denoted│Aij│ - in this case │A11│
 3 5 4
A  6 9 7
2 8 1
9 7
6 7
6 9
A  3
5
4
8 1
2 1
2 8
= 3(9 – 56) - 5(6 – 14) + 4(48 – 18)
= -141 + 40 + 120
= 19
The COFACTOR of the elements of aij denoted by cij
is
cij = (-1)i+j│Aij│
n
n
j 1
i
A   aijcij   aijcij
PROPERTIES OF DETERMINANTS
1.
2.
3.
│AT│=│A│
a11 ka12
a11 a12
 ka 
a21 ka22
a 21 a 22
If A is (n x n) then │kA│ = kn│A│
4.
If a square matrix has two equal rows or columns
its determinant is zero
5.
If any row (or column) is the multiple of any
other row (or column) then its determinant is zero
6.
The value of a determinant is unchanged if a
multiple of one row (or column) is added to
another row (or column)
7.
If A is a diagonal matrix of order n then its
determinant is a11a22 …. ann
8.
If A is a triangular matrix of order n then its
determinant is a11a22 …. ann
9.
If B is the matrix obtained from a square matrix A
by interchanging any two rows (or columns) then
det B = -det A
10. If A and B are square matrices of the same order
then │AB│ = │A││B│
11. If A1, A2 , …. , As are square matrices then
│diag(A1, A2 , …. , As )│ = │A1││A2│ … │As│
12. In general │A + B│does not equal│A│ + │B│
The RANK of a matrix is equal to the highest order nonzero determinant that can be formed from its submatrices
4 5 2 14 
3 9 6 21

A
8 10 7 28


1 2 9 5 
det A = 0
4 5 2
3 9 6  63
8 10 7
Rank of A = 3
The rank of a matrix can also be measured by the
maximum number of linearily independent columns of A
This also equals the maximum number of linearily
independent rows
 4
 5   2
14 
 3
 9  6 
 21
1   2   0   (1)    0
8 
10 7
28
 
   
 
1
2
9
 
   
5
c1a1 + c2a2 + c3a3 + c4a4 = 0
A FULL COLUMN RANK matrix has the same number
of linearily independent columns (rows) equal to the
number of columns
A FULL ROW RANK matrix has the same number of
linearily independent rows (columns) equal to the
number of rows
If A does not have full row and column rank it is
SINGULAR
If A does have full row and column rank it is NONSINGULAR
rank (In) = n
rank (kA) = rank (A)
rank (AT) = rank (A)
If A is (m x n) then rank (A) is ≤ min {m, n}
rank AB ≤ min{rank (A), rank (B)}
INVERSES
If A and B are matrices of order n such that AB = BA = In
then B is called the inverse of A
A has an inverse iff it is of full column and row rank – nonsingular
A-1 = CT / │A│
CT is the transpose of the matrix of co-factors
1 2
A

3 4
│A│= 4 – 6 = -2
c11 = 4 c12 = -3 c21 = -2 c22 = 1
 1  4  3
A 
2  2 1 
T
1
1 
 2
A 

3
/
2

1
/
2


1
PROPERTIES OF INVERSES
1. I-1 = I
2. (A-1)-1 = A
3. AB = I BA = I
4. A non-singular A-1 non-singular
5. A and B non-singular (AB)-1 = B-1A-1
LEFT INVERSE of a (m x n) matrix A is the (n x m)
matrix B such that BA = In
RIGHT INVERSE of a (m x n) matrix A is the (n x m)
matrix C such that AC = Im
(T x k) design matrix X which has rank k < T
has an infinite number of left inverses including
(XTX)-1XT
IDEMPOTENT MATRICES
AA = A
0.4 0.8
A

0.3 0.6
Consider M = [I – X(XTX)-1XT]
KRONECKER PRODUCT
A B
1 3
A

2
0


 2 2 0
B

1
0
3


 2
1
1

A B 
 2
2 
 1
2 0  2
3

0 3  1
2 0  2
0

0 3  1
2
1

4

2
6
0
0
0
2
0
4
0
0
3
0
6
6
3
0
0
0
9
0

0
2 0 

0 3 
2 0 


0 3 