课程名称︰数值线性代数
课程性质︰数学系选修
课程教师︰薛克民
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/11/01
考试时限（分钟）：110
试题 :
Instructions:
‧ Total points 100
‧ Open books, notes, and laptops
‧ Computer software can only be used for numerical validation, but is not used
directly for the solution
‧ Answer the questions thoroughly and justify all your answers
m ×m m T -1
1. (20 points) Let A ∈ |R be nonsingular, u, v ∈ |R , and v A u ≠ -1.
Show that
-1 T -1
T -1 -1 A uv A
(A + uv ) = A - ─────.
T -1
1 + v A u
2. (30 points) Consider the matrix
┌ 1 1 ┐
│ │
A = │ 0 0 │.
│ ＿ ＿│
└ √2 -√2 ┘
Λ
Suppose that we have known Σ and V* of the reduced singular value
decomposition (SVD) of the matrix A as
＿ ＿
^Λ ^┌ 2 0 ┐┌ √2/2 -√2/2 ┐
A = UΣV* = U│ ＿││ ＿ ＿ │.
└ 0 √2┘└ √2/2 √2/2 ┘
^
(a) Compute U.
(b) Construct a full SVD of A = UΣV* for some U and Σ.
(c) What are the four fundamental subspaces of A, i.e. R(A), N(A*), R(A*),
and N(A)?
(d) What is the pseudoinverse of A?
m ×m
3. (25 points) Let Q = Q = [q , q , ..., q ] ∈ |R be a real orthogonal
1 1 2 m
matrix.
T
(a) Determine a reflector P = I - 2u u such that P q = e .
1 1 1 1 1 1
(b) Show that P Q = Q has the form
1 1 2
┌ 1 0 ... 0 ┐
│ │
│ 0 │
Q = │ ~ │
2 │ : Q │
│ 2 │
└ 0 ┘
~ ~ ~ ~ (m-1) ×(m-1)
where Q = [q , q , ..., q ] ∈ |R is a real orthogonal
2 1 2 m-1
matrix.
(c) Using the result in (b), Q can be transformed into a diagonal form with
a sequence of orthogonal transformations
P ...P P .
m-1 2 1
What is this diagonal form?
4. (25 points) Consider the matrix
┌ 3 -3 ┐
│ │
A = │ 0 4 │.
│ │
└ 4 1 ┘
(a) Find the QR factorization of A by Householder reflection.
(b) Use the result in (a) to find the least squares solution of Ax = b, where
T
b = [16 11 17] .
(c) What is the orthogonal projector of A into R(A)?