This exercise explores the connections between the Q method and the matrices in the factorization A = QDQT (see Theorem 4.6). It’s assumed that A is symmetric. (a) In the QR method, show that C1 = QT1...

This exercise explores the connections between the Q
method and the matrices in the factorization
A
=
QDQ

^T

(see Theorem 4.6). It’s assumed that
A
is symmetric.

(a) In the QR method, show that
C
₁
=
Q

^T

₁
AQ
₁,
C
₂
=
Q

^T

₂
Q

^T

₁
AQ
₁
Q
₂, and in general,
C

_k

=
P
^T
_k
AP

_k
, where
P

_k

=
Q
₁
Q
₂
Q

_k
. Note that
P

_k

is an orthogonal matrix (you do not need to show this).

(b) Assuming that
C

_k

converges to a diagonal matrix, explain why the

method is a procedure for computing the matrix
D
in Theorem 4.6. Also explain how two lines of code can be added to the

algorithm given in Section 4.3.3 so that, when finished, you have a matrix containing the eigenvectors.