Spring 2023, CMPSC/MATH 455 Homework Assignment #6
The homework is due April 30th. Questions 1 and 2 are worth 5 points, and 3 and 4 are worth
10 points. Please submit the code (.m files) for problem 4 on Canvas.
1
Show that if the vector v ̸= 0, then the matrix H = I − 2vv
T
vT v
is orthogonal and symmetric.
2
Consider the column vector a as an n×1 matrix. Write out its reduced singular value decomposition,
showing the matrices U , Σ, and V explicitly. Consider the row vector aT as a 1× n matrix. Write
out its reduced SVD, showing the matrices U , Σ, and V explicitly.
3
Suppose that the symmetric matrix B =
[
A a
aT α
]
of order n+1 is positive definite. Show that the
scalar α must be positive and the n× n matrix A must be positive definite. What is the Cholesky
factorization of B in terms of the constituent submatrices?
4
Write an Octave/MATLAB function to compute the standard deviation σ of a sequence xi using
the following two formulas:
The two-pass formula σ =
[
1
n− 1
n∑
i=1
(xi − x¯)2
]1/2
, and
the one-pass formula σ =
[
1
n− 1
(
(
n∑
i=1
x2i )− nx¯2
)]1/2
.
Your function should accept a vector x as input and return the values computed using the above
formulas as output. You can use the Octave/MATLAB built-in function to compute the mean x¯.
(a) List the program. Report the output for an input sequence of your choice.
(b) Can you devise an input data sequence that illustrates the numerical difference between the
two mathematically equivalent formulas?