Fin500J%20Mathematical%20Foundations%20in%20Finance

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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

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Outline

• Definition of a Matrix
• Operations of Matrices
• Determinants
• Inverse of a Matrix
• Linear System
• Matrix Definiteness

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Matrix (Basic Definitions)
An k n matrix A is a rectangular array of
numbers with k rows and n columns. (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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Operations with Matrices (Sum, Difference)
Sum, Difference If 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
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Operations with Matrices (Scalar Multiple)
Scalar Multiple If 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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Operations with Matrices (Product)
Product If A has dimensions k m and B has
dimensions m n, then the product AB is defined,
and has dimensions k n. The entry (AB)ij is
obtained by multiplying row i of A by column j of
B, which is done by multiplying corresponding
entries together and then adding the results i.e.,
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Laws of Matrix Algebra

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

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Operations with Matrices (Transpose)
Transpose The transpose, AT , of a matrix A is
the matrix obtained from A by writing its rows as
columns. If A is an kn matrix and B AT then B
is the nk matrix with bij aji. If ATA, then A
is symmetric.
  Example
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Determinants

• Determinant 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

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

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Determinants

• The determinant of a 2 2 matrix A is

• The determinant of a 3 3 matrix is

• Example

• In Matlab det(A) det(A)

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Inverse of a Matrix

• Definition. If A is a square matrix, i.e., A has
  dimensions nn. Matrix A is nonsingular or
  invertible if there exists a matrix B such that
  ABBAIn. For example.

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

• 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)-1A.

• Matrix division
• A/B AB-1

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

• In Matlab A-1 inv(A)

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Calculation of Inversion using Determinants
Def For any nn matrix A, let Cij denote the
(i,j) th cofactor of A, that is, (-1)ij times
the determinant of the submatrix obtained by
deleting row i and column j form A, i.e., Cij
(-1)ij Mij . The nn 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
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Calculation of Inversion using Determinants
Example find the inverse of the matrix Solve
thus
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

• Elementary row operations
• Interchange two rows of a matrix
• Change a row by adding to it a multiple of
  another row
• 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
is then the matrix inverse of A
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Example
(ii)(-12)(i), (iii)(-3) (i), (iii)(ii)
(1/10)
The matrix
is then the matrix inverse of A
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Systems of Equations in Matrix Form
The system of linear equations
can be rewritten as the matrix equation Axb,
where
If an nn matrix A is invertible, then it is
nonsingular, and the unique solution to the
system of linear equations Axb is xA-1b.
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Example solve the linear system

• In Matlab
• gtgtxinv(A)b
• or
• gtgt xA\b

b
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Matrix Operations in Matlab
gtgt A2 3 1 1 1 0 A 2 3 1
1 1 0 gtgt B11 1 0 1 2 4 B1 1
1 0 1 2 4 gtgt B21 1 1 1 0
2 B2 1 1 1 1 0 2
gtgt AB1 ans 3 4 1 2 3
4 gtgt A-B1 ans 1 2 1 0 -1
-4 gtgt AB2 ans 5 2 8 2
1 3 1 1 1
Sum Difference Product
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Matrix Operations in Matlab
gtgt C' ans 1 12 3 1 2
4 1 -3 1 gtgt det(C) ans 35 gtgt
inv(C) ans 0.4000 0.0857 -0.1429
-0.6000 -0.0571 0.4286 1.2000 -0.0286
-0.2857
transpose determinant inverse
gtgt C1 1 1 12 2 -3 3 4 1 C 1 1
1 12 2 -3 3 4 1
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Positive Definite Matrix
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Negative Definite, Positive Semidefinite,
Negative Semidefinite, Indefinite Matrix

• Let A be an NN symmetric matrix, then A is
• negative definite if and only if vTAv lt0 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 gt0 for some v in
  RN and lt0 for other v in RN