18.335 Midterm
You have two hours.
All problems have equal weight
, but some problems may be harder than others, so try
not to spend too much time on one problem.
Problem 1: Schur, backsubstitution, complexity (20 points)
You are given matrices
A
(
m
×
m
),
B
(
n
×
n
), and
C
(
m
×
n
), and want to solve for an unknown
matrix X
(
m
×
n
) solving:
AX

XB
=
C
.
We will do this using the Schur decompositions of
A
and
B
. (Recall that any square matrix
S
can be factorized
in the Schur form
S
=
QUQ
*
for some unitary matrix
Q
and some uppertriangular matrix
U
.)
(a)
Given
the Schur decompositions
A
=
Q
A
U
A
Q
*
A
and
B
=
Q
B
U
B
Q
*
B
, show how to transform
AX

XB
=
C
into new equations
A
0
X
0

X
0
B
0
=
C
0
, where
A
0
and
B
0
are upper triangular and
X
0
is the new
m
×
n
matrix of unknowns. That is, express,
A
0
,
B
0
, and
C
0
in terms of
A
,
B
, and
C
(or their Schur factors),
and show how to get
X
back from
X
0
.
(b) Given the uppertriangular system
A
0
X
0

X
0
B
0
=
C
0
from (a), give an algorithm to ﬁnd the
last row
of
X
0
. (Hint: look at the last row of the equation.)
(c) Suppose you have computed all rows
>
j
of
X
0
, give an algorithm to compute the
j
th row.
(d) The combination of the previous three parts yields a backsubstitutionlike algorithm to solve for
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 Spring '06
 Prof.CarolLivermore
 Matrices, Singular value decomposition, Hermitian matrix, SVDs, Schur decompositions, Schur factors

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