Solution to Homework II
Problem 6.1
(1) To prove that I 2P is unitary, we show that
(I 2P ) (I 2P ) = I 4(P + P ) + 4P P = I,
by using the denition of P being an orthogonal projector
P = P.
P 2 = P,
(2) Let H be the hyperplane orthogonal to Q = rang(P ).
Solution to Homework I
Problem 1.3
It is easy to prove that
(1) An square upper-triangular matrix is non-singular if and only if all its
diagonal elements are non-zero.
(2) An m m matrix R is a non-singular upper-triangular matrix if and
only if
span(e1 ,
Solution to Homework IV
Problem 20.1
1.
Assume that A has an LU factorization A = LU and write it in
the block structure
A=
A11 A12
A21 A22
=
L11 0
L21 L22
U11 U12
0 U22
.
Then, the upper-left k k block A11 also has an LU factorization:
A11 = L11 U11 ,
i
Solution to Homework II
Problem 6.1
(1) To prove that I 2P is unitary, we show that
(I 2P ) (I 2P ) = I 4(P + P ) + 4P P = I,
by using the denition of P being an orthogonal projector
P = P.
P 2 = P,
(2) Let H be the hyperplane orthogonal to Q = rang(P ).
Solution to Homework I
Problem 1.3
It is easy to prove that
(1) An square upper-triangular matrix is non-singular if and only if all its
diagonal elements are non-zero.
(2) An m m matrix R is a non-singular upper-triangular matrix if and
only if
span(e1 ,
Solution to Homework IV
Problem 20.1
1.
Assume that A has an LU factorization A = LU and write it in
the block structure
A=
A11 A12
A21 A22
=
L11 0
L21 L22
U11 U12
0 U22
.
Then, the upper-left k k block A11 also has an LU factorization:
A11 = L11 U11 ,
i