Problems 1.* are from the Review on p.60, prblems 1.x.* are from section x
in chapter 1.
Part A: Strang 1.4.15, 1.5.13, 1.1, 1.14, 1.19, 1.27, 1.29
Part B: Strang 1.3.11, 1.4.14, 1.4.17, 1.6.19, 1.6.23 and the following 2
Is it true that for a pair of elementary permutation and elementary
lower triangular matrices L and P there exists another pair of such
matrices L' and P' so that
L' P' = P L ?
a) Check whether it is true in general.
b) Check whether it is true if
* L corresponds to an elimination step in the Gaussian method,
* P corresponds to a permutation step in the Gaussian method,
* the elimination step precedes the permutation step.
Note: we do not assume here that the permutation step follows
IMMEDIATELY after the elimination step in the Gaussian method,
there might be other steps between them.
a) Prove that a product of two lower triangular matrices L' and L''
is again a lower triangular matrix
L' L'' = L
b) Prove that a product of two permutation matrices P' and P''
is again a permutation matrix
P' P'' = P