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C&O 367, Winter 2001
Assignment 2
Due on Thursday, Jan. 25, (at start of class)
Instructor H. Wolkowicz
- (10 marks)
(Text: Problem 31, page 36)
- Let
be an
symmetric matrix. Diagonalize
to show
that (the Raleigh quotient)
is greater than or equal to the smallest eigenvalue of
for all
in
.
- Show that the quadratic form
is coercive if and only if
is positive definite.
- Conclude from 1b that if
is any quadratic function where
and
is an
symmetric matrix, then
is coercive if and only if
is positive definite.
- (5 marks)
Suppose that
where
and
is an
symmetric matrix.
Show that
is bounded below on
if and only if the minimum
of f on
is attained (i.e. there exists
such that
).
- (5 marks)
(Text: Problems 1a and 1d, page 77)
- (5 marks)
(Text: Problems 2b and 2c, page 77)
- (5 marks)
(Text: Problems 9, page 78)
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Henry Wolkowicz
2001-01-22