Fin500J Mathematical Foundations in Finance Topic 1: Matrix Algebra Philip H. Dybvig Reference:...

Fin500J Mathematical Foundations in Finance

Topic 1: Matrix Algebra Philip H. Dybvig

Reference: Mathematics for Economists, Carl Simon and Lawrence Blume, Chapter 8 Chapter 9 and Chapter 16

Slides designed by Yajun Wang

1Fall 2010 Olin Business SchoolFin500J Topic 1

OutlineDefinition of a MatrixOperations of MatricesDeterminantsInverse of a MatrixLinear SystemMatrix Definiteness

Fall 2010 Olin Business School 2Fin500J Topic 1

ij

knk

n

n

A

aa

aa

aa

,,

,,

,,

1

221

111

A

An k × n matrix A is a rectangular array of numbers with k rows and ncolumns. (Rows are horizontal and columns are vertical.) The numbers k and n are the dimensions of A. The numbers in the matrix are called its entries. The entry in row i and column j is called aij .


Sum, DifferenceIf A and B have the same dimensions, then their sum, A + B, is obtained by adding corresponding entries. In symbols, (A + B)ij = aij + bij . If A and B have the same dimensions, then their difference, A − B, is obtained by subtracting corresponding entries. In symbols, (A - B)ij = aij - bij .

Example:

4Fall 2010 Olin Business School

AA

0

A allfor Then, zero. all are entries whose0matrix The

1 12 12

84 2

1 5 6

70 1

0 7 6

14 3

Fin500J Topic 1

Scalar MultipleIf A is a matrix and r is a number (sometimes called a scalar in this context), then the scalar multiple, rA, is obtained by multiplying every entry in A by r. In symbols, (rA)ij = raij .

Example:


0 14 12

28 6

0 7 6

14 32

Fin500J Topic 1

ProductIf A has dimensions k × m and B has dimensions m × n, then the productAB is defined, and has dimensions k × n. The entry (AB)ij is obtainedby multiplying row i of A by column j of B, which is done by multiplyingcorresponding entries together and then adding the results i.e.,


B.IB B,matrix mnany

for andA AI A,matrix n many for

100

01 0

00 1

Imatrix Identity

.

Example

....)...( 22112

1

21

nn

mjimjiji

mj

j

j

imii

fDeBfCeA

dDcBdCcA

bDaBbCaA

DC

BA

fe

dc

ba

bababa

b

b

b

aaa

Fin500J Topic 1

Laws of Matrix Algebra

The matrix addition, subtraction, scalar multiplication and matrix multiplication, have the following properties.

Fall 2010 Olin Business School 7

BC. AC B)C AC, (A AB C) A(B

A B B A

A(BC). C, (AB)C B) (A C) (B A

:Laws veDistributi

:Additionfor Law eCommutativ

:Laws eAssociativ

Fin500J Topic 1

TransposeThe transpose, AT , of a matrix A is the matrix obtained from A by writing its rows as columns. If A is an k×n matrix and B = AT then B is the n×k matrix with bij = aji. If AT=A, then A is symmetric.

Example:


,CD(CD)

rA(rA)A,)(A

,BAB)(A,BAB)(A

aa

aa

aa

aaa

aaa

TTT

TTTT

TTTTTT

T

Then, matrix.n man be D andmatrix mk a be CLet

scalar. a isr andn k are B andA where

: verifyeasy toit It

2313

2212

2111

232221

131211

Fin500J Topic 1

DeterminantsDeterminant is a scalar

Defined for a square matrix Is the sum of selected products of the elements of the matrix,

each product being multiplied by +1 or -1

11 12 1

21 22 2

1 1

1 2

det( ) ( 1) ( 1)

n

n nn i j i j

ij ij ij ijj i

n n nn

a a a

a a aA a M a M

a a a


• Mij=det(Aij), Aij is the (n-1)×(n-1) submatrix obtained by deleting row i and column j from A.

Fin500J Topic 1

Determinants

The determinant of a 3 ×3 matrix is

11 12 1322 23 21 23 21 221 1 1 2 1 3

21 22 23 11 12 1331 3232 33 31 33

31 32 33

( 1) ( 1) ( 1)

a a aa a a a a a

a a a a a aa aa a a a

a a a

Example

10

1 1 1 2 1 3

1 2 35 6 4 6 4 5

4 5 6 1( 1) 2( 1) 3( 1)8 10 7 10 7 8

7 8 10

50 48 2(40 42) 3(32 35) 3

10

bcaddc

baA )det(The determinant of a 2 ×2 matrix A

is

• In Matlab: det(A) = det(A)

Fall 2010 Olin Business SchoolFin500J Topic 1

Inverse of a MatrixDefinition. If A is a square matrix, i.e., A has dimensions n×n.

Matrix A is nonsingular or invertible if there exists a matrix B such that AB=BA=In. For example.

Common notation for the inverse of a matrix A is A-1

If A is an invertible matrix, then (AT)-1 = (A-1)T

The inverse matrix A-1 is unique when it exists. If A is invertible, A-1 is also invertible A is the inverse matrix of A-1. (A-1)-1=A.

• In Matlab: A-1 = inv(A)

• Matrix division:

A/B = AB-1


10

01

3

2

3

1

3

2

3

23

1

3

1

3

1

3

2

3

1

3

13

1

3

2

21

11

Fin500J Topic 1

Calculation of Inversion using Determinants

Def: For any n×n matrix A, let Cij denote the (i,j) th cofactor of A, that is, (-1)i+j times the determinant of the submatrix obtained by deleting row i and column j form A, i.e., Cij = (-1)i+j Mij . The n×n matrix whose (i,j)th entry is Cji, the (j,i)th cofactor of A is called the adjoint of A and is written adj A.

thus

-1

Thm: Let A be a nonsingular matrix. Then,

1A .

detadj A

A


Calculation of Inversion using Determinants

thus

Example: find the inverse of the matrix

Solve:

2 4 5

0 3 0

1 0 1

A

11 12 13

21 22 23

31 32 33

11 21 31

12 22 32

13 23 33

1

3 0 0 0 0 33, 0, 3,

0 1 1 1 1 0

4 5 2 5 2 44, 3, 4,

0 1 1 1 1 0

4 5 2 5 2 415, 0, 6,

3 0 0 0 0 3

det 9,

3 4 15

0 3 0 .

3 4 6

31

,9

C C C

C C C

C C C

A

C C C

adjA C C C

C C C

So A

4 15

0 3 0 .

3 4 6

13

Using Determinants to find the inverse of a matrix can be very complicated. Gaussian elimination is more efficient for high dimension matrix.


Calculation of Inversion using Gaussian Elimination

14

Elementary row operations: o Interchange two rows of a matrixo Change a row by adding to it a multiple of another rowo Multiply each element in a row by the same nonzero number

• To calculate the inverse of matrix A, we apply the elementary row operations on the augmented matrix [A I] and reduce this matrix to the form of [I B]

• The right half of this augmented matrix B is the inverse of A


Calculation of inversion using Gaussian elimination

I is the identity matrix, and use Gaussian elimination to obtain a matrix of the form

The matrix

100 ,,

01 0 ,,

00 1 ,,

][

1

221

111

nnn

n

n

aa

aa

aa

IA


nnn

n

n

aa

aa

aa

,,

,,

,,

1

221

111

A

nnnn

n

n

bbb

bbb

bbb

100

01 0

00 1

21

22221

11211

nnnn

n

n

bbb

bbb

bbb

B

21

22221

11211

is then the matrix inverse of A

Fin500J Topic 1

Example

The matrix

1 1 1 |1 0 0

[ | ] 12 2 3 | 0 1 0

3 4 1 | 0 0 1

A I

16

1 1 1

12 2 3

3 4 1

A

is then the matrix inverse of A

1 1 1 | 1 0 0

0 10 15 | 12 1 0

0 0 3.5 | 4.2 0.1 1

3 11 0 0 | 0.4

35 72 3

0 1 0 | 0.635 71 2

0 0 1 | 1.235 7

(ii)+(-12)×(i), (iii)+(-3) ×(i), (iii)+(ii) ×(1/10)

3 10.4

35 72 3

0.635 71 2

1.235 7


The system of linear equations


Systems of Equations in Matrix Form11 1 12 2 13 3 1 1

21 1 22 2 23 3 2 2

1 1 2 2 3 3

n n

n n

k k k kn n k

a x a x a x a x b

a x a x a x a x b

a x a x a x a x b

can be rewritten as the matrix equation Ax=b, where

1 111 1

2 2

1

, , .n

k knn k

x ba a

x bA x b

a ax b

If an n×n matrix A is invertible, then it is nonsingular, and the unique solution to the system of linear equations Ax=b is x=A-1b.

Fin500J Topic 1

Example: solve the linear system

1

-1

Matrix Inversion

4 1 2 x 4

5 2 1 ; X y ; b 4

1 0 3 z 3

6 -3 -31

A -14 10 66

-2 1 3

x 6 -3 -3 41

y -14 10 6 46

z -2 1 3 3

1 2; y 1 3; z 5 6

AX d

A

X A b

x

18

4 2 4

5 2 4

3 3

x y z

x y z

x z

Fall 2010 Olin Business School

• In Matlab >>x=inv(A)*b or >> x=A\b

b

Fin500J Topic 1


Matrix Operations in Matlab

>> A=[2 3; 1 1; 1 0]

A =

2 3

1 1

1 0

>> B1=[1 1; 0 1; 2 4]

B1 =

1 1

0 1

2 4

>> B2=[1 1 1; 1 0 2]

B2 =

1 1 1

1 0 2

>> A=[2 3; 1 1; 1 0]

A =

2 3

1 1

1 0

>> B1=[1 1; 0 1; 2 4]

B1 =

1 1

0 1

2 4

>> B2=[1 1 1; 1 0 2]

B2 =

1 1 1

1 0 2

>> A+B1

ans =

3 4

1 2

3 4

>> A-B1

ans =

1 2

1 0

-1 -4

>> A*B2

ans =

5 2 8

2 1 3

1 1 1

>> A+B1

ans =

3 4

1 2

3 4

>> A-B1

ans =

1 2

1 0

-1 -4

>> A*B2

ans =

5 2 8

2 1 3

1 1 1

Sum

Difference

Product

Fin500J Topic 1


Matrix Operations in Matlab

>> C=[1 1 1; 12 2 -3; 3 4 1]

C =

1 1 1

12 2 -3

3 4 1

>> C=[1 1 1; 12 2 -3; 3 4 1]

C =

1 1 1

12 2 -3

3 4 1

>> C'

ans =

1 12 3

1 2 4

1 -3 1

>> det(C)

ans =

35

>> inv(C)

ans =

0.4000 0.0857 -0.1429

-0.6000 -0.0571 0.4286

1.2000 -0.0286 -0.2857

>> C'

ans =

1 12 3

1 2 4

1 -3 1

>> det(C)

ans =

35

>> inv(C)

ans =

0.4000 0.0857 -0.1429

-0.6000 -0.0571 0.4286

1.2000 -0.0286 -0.2857

transpose

determinant

inverse

Fin500J Topic 1

Fall 2010 Olin Business School 21Fin500J Topic 1

Fall 2010 Olin Business School 22

Let A be an N×N symmetric matrix, then A is

• negative definite if and only if vTAv <0 for all v≠0 in RN

• positive semidefinite if and only if vTAv ≥0 for all v≠0, in RN

• negative semidefinite if and only if vTAv ≤0 for all v≠0, in RN

• indefinite if and only if vTAv >0 for some v in RN and <0 for other v in RN

Fin500J Topic 1

Fin500J Mathematical Foundations in Finance Topic 1: Matrix Algebra Philip H. Dybvig Reference:...

Documents

Transcript of Fin500J Mathematical Foundations in Finance Topic 1: Matrix Algebra Philip H. Dybvig Reference:...