Page 167 - 35Linear Algebra

7.8 Review Problems 167

k
(k) Try to find a formula or recursive method for finding x . Don’t
worry about simplifying your answer.

X Y
2. Let M = be a square n × n block matrix with W invertible.
Z W
i. If W has r rows, what size are X, Y , and Z?
ii. Find a UDL decomposition for M. In other words, fill in the stars
in the following equation:

X Y I ∗ ∗ 0 I 0
=
Z W 0 I 0 ∗ ∗ I
3. Show that if M is a square matrix which is not invertible then either
the matrix matrix U or the matrix L in the LU-decomposition M = LU
has a zero on it’s diagonal.

4. Describe what upper and lower triangular matrices do to the unit hy-
percube in their domain.

5. In chapter 3 we saw that, since in general row exchange matrices are
necessary to achieve upper triangular form, LDPU factorization is the
complete decomposition of an invertible matrix into EROs of various
kinds. Suggest a procedure for using LDPU decompositions to solve
linear systems that generalizes the procedure above.

6. Is there a reason to prefer LU decomposition to UL decomposition, or
is the order just a convention?

7. If M is invertible then what are the LU, LDU, and LDPU decompo-
sitions of M T in terms of the decompositions for M? Can you do the
same for M −1 ?

T
8. Argue that if M is symmetric then L = U in the LDU decomposition
of M.

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