Written Assignment #2
> restart;
> with(LinearAlgebra):
Problem 1
Work this problem by hand and turn in on a separate piece of paper.
Let be the matrix
Find
by hand computation (Hint: you're looking at the Jordan Decomposition).
In each of the following problems, do the following:
1 Find the matrices
commutes with
and in the Jordan Decomposition
and that is nilpotent. Is diagonalizable?
2 Find
. Don't worry if
,
, and
Verify the your
is to big to display nicely.
3 For the given initial condition , solve the vector initial value problem
. Check you answer by differentiation.
Problem 2
> A := Matrix(3, 3, {(1, 1) = 5, (1, 2) = -1, (1, 3) = 3, (2, 1) =
9, (2, 2) = -5, (2, 3) = 9, (3, 1) = 2, (3, 2) = -3, (3, 3) = 4})
;
(1)
> c := Vector(3, {(1) = 1, (2) = 2, (3) = -1});
(2)
>
Problem 3
> A := Matrix(3, 3, {(1, 1) = 51, (1, 2) = -106, (1, 3) = -176, (2,
1) = 18, (2, 2) = -38, (2, 3) = -60, (3, 1) = 4, (3, 2) = -8, (3,
3) = -15});
(3)
> c := Vector(3, {(1) = 1, (2) = 2, (3) = -1});
(4)
Problem 4
> A := Matrix(6, 6, {(1, 1) = -1101, (1, 2) = 179, (1, 3) = -305,
(1, 4) = -1079, (1, 5) = 660, (1, 6) = 1124, (2, 1) = 3008, (2,
2) = -540, (2, 3) = 845, (2, 4) = 3231, (2, 5) = -2027, (2, 6) =
-3196, (3, 1) = 5453, (3, 2) = -910, (3, 3) = 1516, (3, 4) =
5467, (3, 5) = -3351, (3, 6) = -5612, (4, 1) = 387, (4, 2) = -75,
(4, 3) = 110, (4, 4) = 446, (4, 5) = -283, (4, 6) = -424, (5, 1)
= -107, (5, 2) = 19, (5, 3) = -30, (5, 4) = -113, (5, 5) = 70,
(5, 6) = 112, (6, 1) = 352, (6, 2) = -68, (6, 3) = 100, (6, 4) =
402, (6, 5) = -250, (6, 6) = -381});
(5)
> c := Vector(6, {(1) = 1, (2) = -1, (3) = 0, (4) = 2, (5) = -3,
(6) = 0});
(6)
(6)
Problem 5
> A := Matrix(7, 7, {(1, 1) = -37, (1, 2) = 115, (1, 3) = -70, (1,
4) = -9, (1, 5) = 66, (1, 6) = 35, (1, 7) = -35, (2, 1) = -205,
(2, 2) = 145, (2, 3) = -192, (2, 4) = -8, (2, 5) = -29, (2, 6) =
99, (2, 7) = -53, (3, 1) = 21, (3, 2) = 18, (3, 3) = -2, (3, 4) =
0, (3, 5) = 11, (3, 6) = 0, (3, 7) = -9, (4, 1) = -1064, (4, 2) =
775, (4, 3) = -1012, (4, 4) = -42, (4, 5) = -157, (4, 6) = 521,
(4, 7) = -285, (5, 1) = -3, (5, 2) = 6, (5, 3) = -6, (5, 4) = 0,
(5, 5) = -2, (5, 6) = 3, (5, 7) = -3, (6, 1) = -79, (6, 2) = 428,
(6, 3) = -241, (6, 4) = -30, (6, 5) = 239, (6, 6) = 119, (6, 7) =
-139, (7, 1) = -496, (7, 2) = 571, (7, 3) = -555, (7, 4) = -37,
(7, 5) = 96, (7, 6) = 284, (7, 7) = -194});
(7)
> c := Vector(7, {(1) = 1, (2) = 0, (3) = 0, (4) = -2, (5) = 0, (6)
= 1, (7) = 1});
(8)