Partial Solutions to Homework II
Y. Zhou
Section 4.2 5.
Proof. We are going to use the mathematical induction and the conclusion from 1.a which says that
the inverse of a upper triangular matrix is also a upper triangular.
We first prove that nonsingular upper triangular n × n matrix have all diagonal elements nonzero.
This is obviously true for n = 1. Suppose it is also true for n = k−1, i.e., a nonsingular (k−1)×(k−1)
upper triangular matrix Uk−1 has all diagonal elements nonzero. Then it is sufficient to show that
the n × n nonsingular matrix Uk of the form
Uk−1 V
0
a
with arbitrary vector V shall have a 6= 0. Since Uk is nonsingular, we can show that its inverse is of
the form
(Uk−1 )−1 W
0
1/a
for appropriate vector W . Thus a can not be zero otherwise Uk is irreversible.
To prove the converse statement we use the mathematical induction also. For n = 1 the element
is nonzero directly indicates the existence of the inverse. Now we assume that all elements of Uk−1
are nonzero and Uk−1 is invertible. But then it is sufficient to prove that the matrix
Uk−1 V
0
a
with a 6= 0 is also invertible. Since we can explicitly find the inverse of this n × n matrix, as did
above, the nonsingularity of this matrix follows.
The statement about the lower triangular matrix can be proved similarly.
13. Suppose that A has an LU decomposition, i.e., there are L and U such that
0 0
u11 u12
l11 0
=
.
×
a b
0 u22
l21 l22
It can be found that l11 = 0, and l21 , l22 , u11 , u12 , u22 are free to choose as long as
l21 u11 = a,
l21 u12 + l22 u22 = b.
Thus A has LU decomposition. And it is impossible for L to be a unit lower triangular matrix since
l11 is zero anyway.
14. Can be shown following the similar arguments for 13.
36 Note that in Doolittle’s LU decomposition we have L with unit diagonal, i.e.,
A = LU.
In order to have an lower triangular matrix with 2’s on its main diagonal, we decompose U as the
product of a matrix D with 2’s on its main diagonal and zeros elsewhere, and a matrix Û :
A = LU = LDÛ = L̂Û .
Since D is a diagonal matrix, L̂ will still be a lower triangular matrix. For the decomposition of U
into DÛ please refer Example 1 in page 155 of the textbook.
1