fin500j mathematical foundations in finance topic 1: matrix algebra philip h. dybvig reference:...
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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 .
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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:
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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.,
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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:
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,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
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• 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
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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
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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.
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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
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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
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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
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The system of linear equations
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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
19Fall 2010 Olin Business School
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
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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
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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