Mathematical Methods for Quantitative Finance Homework 4

CFRM 405: Mathematical Methods for Quantitative Finance Homework 4
1. Let A =


1 1 0
1 4 1
−2 1 1


(a) Use elimination to turn A into an upper triangular matrix. How many pivots does A
have?
(b) Let b = (1, 6, 3). Does Ax = b have a solution?
(c) Let b = (1, 6, 5). Does Ax = b have a solution?
(d) Can you find multiple solutions in either part (b) or part (c)? If so, find 2.
(e) Does A have an inverse? Justify your answer using results from this exercise.
2. Suppose AB = I and CA = I where I is the n × n identity matrix.
(a) What are the dimensions of the matrices A, B and C?
(b) Show that B = C.
[Hint: you can write IB = B]
(c) Is A invertible?
3. Let A be a square matrix with the property that A2 = A. Simplify (I − A)
2 and (I − A)
7
.
4. (a) Write the vector (9, 2, −5) as a linear combination of the vectors (1, 2, 3) and (6, 4, 2)
or explain why it can’t be done.
(b) How many pivots does a system of equations with coefficient matrix
A =


1 6 9
2 4 2
3 2 −5


have?
5. Suppose A is a 6 × 20 matrix and B is a 20 × 7 matrix.
(a) What are the dimensions of C = AB?
(b) Suppose A, B, and C have been partitioned into block matrices like so:
A =

A11 A12 A13
A21 A22 A23 
, B =


B11 B12
B21 B22
B31 B32

 , C =

C11 C12
C21 C22 
,
Suppose that A11 is 2 × 10, B22 is 4 × 3, and C11 is ? × 4. What are the dimensions
of each block of A, B, and C such that all the resulting block matrix multiplications
are valid?
[Hint: Make note of every fact you know, sketch all three matrices, and fill in the
unknowns step by step]
(c) Write each block of C in terms of blocks of A and B.
6. Let A be an m × n matrix.
(a) The full A = QR factorization contains more information than necessary to reconstruct A. What are the smallest matrices Q˜ and R˜ such that Q˜R˜ = A?
(b) Let A˜ be an m × n matrix (m > n) whose columns each sum to zero, and let A˜ = Q˜R˜
be the reduced QR factorization of A˜. The squared Mahalanobis distance to the point
x˜
T
i
(the i
th row of A˜) is
d
2
i = ˜x
T
i Sˆ−1x˜i
where Sˆ =
1
m−1A˜TA˜ is a covariance matrix. Compute d
2
i without inverting a matrix.
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