Page 109 - Matrix Analysis & Applied Linear Algebra

3.5 Matrix Multiplication 103

3.5.5. Suppose that A and B are m × n matrices. If Ax = Bx holds for
all n × 1 columns x, prove that A = B. Hint: What happens when
x is a unit column?

1/2 α n
3.5.6. For A = , determine lim n→∞ A . Hint: Compute a few
0 1/2
n
powers of A and try to deduce the general form of A .
3.5.7. If C m×1 and R 1×n are matrices consisting of a single column and
a single row, respectively, then the matrix product P m×n = CR is
sometimes called the outer product of C with R. For conformable
matrices A and B, explain how to write the product AB as a sum of
outer products involving the columns of A and the rows of B.

3.5.8. A square matrix U =[u ij ] is said to be upper triangular whenever
u ij = 0 for i>j —i.e., all entries below the main diagonal are 0.
(a) If A and B are two n × n upper-triangular matrices, explain
why the product AB must also be upper triangular.
(b) If A n×n and B n×n are upper triangular, what are the diagonal
entries of AB?
(c) L is lower triangular when ' ij = 0 for i<j. Is it true that
the product of two n × n lower-triangular matrices is again
lower triangular?

3.5.9. If A =[a ij (t)] is a matrix whose entries are functions of a variable t,
the derivative of A with respect to t is defined to be the matrix of
derivatives. That is,

dA da ij
= .
dt dt
Derive the product rule for differentiation

d(AB) dA dB
= B + A .
dt dt dt

3.5.10. Let C n×n be the connectivity matrix associated with a network of n
nodes such as that described in Example 3.5.2, and let e be the n × 1
column of all 1’s. In terms of the network, describe the entries in each
of the following products.
(a) Interpret the product Ce.
T
(b) Interpret the product e C.