1. MF-852 Financial Econometrics
Lecture 1
Overview of Matrix Algebra and Excel
Roy J. Epstein
Fall 2003

2. Contact Info BC

3. Matrices—Some Definitions
A matrix is a rectangular array of real numbers.
Denote a matrix with an upper case italic letter, e.g.,

4. Matrices—Some Definitions
A has m rows and n columns.
The dimension of A is “m x n” (m by n).
A vector is a matrix with a single row or a single column (denoted with a lower case letter without a subscript), e.g,

5. Matrices—Some Definitions
If n = m then A is a square matrix.
If A is square and aij = aji then A is symmetric, e.g.,
is a symmetric matrix.

6. Scalar Multiplication
Let k be a scalar (i.e., a given real number). kA is a new matrix where each element equals kaij.

7. Matrix Addition
A + B is a new matrix where each element is the sum of the corresponding elements of A and B.

8. Matrix Addition
Addition is only defined if the matrices have the same dimensions.
makes no sense.

9. Matrix Subtraction
A – B = A + (–1)B. So subtraction also requires the matrices to have the same dimensions.

10. Matrix Transpose
The transpose of a matrix is a new matrix where the rows and columns are switched. The transpose of A is denoted A .

11. Matrix Rules for Addition
A + B = B + A
(A + B) + C = A + (B + C)
(A + B)  = A + B 

12. Summation Operator
We often need to add up the elements of a vector or matrix. We use a convenient notation for this.
Suppose a = (a1, a2, …, an). Define

13. “Dot” (or “Inner”) Product
The “dot” product “multiplies” two vectors (we ignore other vector multiplication concepts).
The dot product of a and b is defined as ab = ∑aibi (a and b must have the same number of elements).
Suppose
then ab = 1 x x 2 + 2(-3) = 2.

14. Matrix Multiplication
Matrix multiplication AB is defined in terms of dot products. The result is a new matrix C.
Each cij is a dot product.
The dot product involves the ith row of A and the jth column of B.

15. Matrix Multiplication
Example:

16. Matrix Multiplication
# columns in A = # rows in B or matrix multiplication is not defined.
Questions:
Does AB = BA?
Do A and B have to be square in order to multiply them?
Is AA square?

17. Matrix Rules for Multiplication
A(BC ) = (AB)C
A(B+C) = AB + AC
(B+C)A = BA + CA
(AB) = B A
(ABC) = C BA

18. The Identity Matrix
An identity matrix has 1’s on the diagonal and 0 everywhere else.
aii = 1, aij=0 i  j
is a 3 x 3 identity matrix.

19. The Identity Matrix and Multiplication
For scalars, 1 is the multiplicative identity: 1() = ()1 = .
For matrices, AI = IA = A.

20. Matrix Inverse
For   0, –1 is the multiplicative inverse: –1() = ()–1 = 1.
For matrices, A –1 is the inverse of A: A –1A = A A –1 = I.
Excel (and other programs) will calculate A –1.

21. Inverse Matrix Example

22. Singular Matrix For  = 0, –1 does not exist.
Sometimes A –1 does not exist. In this case A is called singular or non-invertible.
Excel sometimes calculates an inverse of a singular matrix when there is a lot of roundoff error—be aware!

23. Singular and Non-Singular Matrices

24. Block Diagonal Matrix
Suppose A has sub-matrices along its main diagonal and is zero elsewhere. Then A is block diagonal, e.g.,

25. Matrices and a System of Linear Equations
Suppose we have a system of 2 linear equations in 2 unknowns, e.g.,
Matrices lead to a simple solution.

26. Matrix Representation
The equation coefficients are a matrix A:
The unknowns are a vector x:
The constants are a vector c.

27. Matrix Solution The equation system can be written
Multiply both sides by A –1
Solution is (since A –1A =I)

28. Matrix Solution
Check:
So x1 = 0 and x2 = 0.5.

29. Excel and Matrices SUMPRODUCT MMULT TRANSPOSE MINVERSE Dot product
Matrix and vector multiplication
TRANSPOSE
Matrix and vector transpose
MINVERSE
Matrix inverse

30. Excel and Matrices
Highlight a range of cells that has the proper dimension for the result of the matrix function.
Enter the matrix formula, e.g., =MINVERSE(b3:d5)
Press CTRL+SHIFT+ENTER.