Problem set 4
• For each of the following matrices (of order n)
– determine eigenvalues and eigenvectors;
– identify the directions defined by eigenvectors (eigendirections); when eigenvectors
are real and n = 2, provide their graphical representation in the cartesian plane;
– say if the matrix can be diagonalized;
– when possible, determine the invertible matrix S (with eigenvectors of A as columns)
and the diagonal matrix Λ such that Λ = S −1 AS; by direct computation, verify that
Λ = S −1 AS and A = SΛS −1
7 −4
i) A =
5 −2
5 12 −6
ii) A = −3 −10 6
−3 −12 5
1 1
iii) A =
0 1
−2 −6
iv) A =
3 4
7 4
v) A =
−1 3
−2 1 0
vi) A = 0 −1 −1
0 −3 −3
2 0 −1
vii) A = 4 1 −4
2 0 −1
0 2 2
viii) A = 2 0 2
2 2 0
0 0 1
ix) A = 1 0 −1
0 1 1
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−2 1 4
x) A = −2 1 4
−2 1 4
• Given the homogeneous linear system of the first order
ẋ1 −1 0 x1
=
ẋ2 0 −3 x2
1. find fundamental solutions (those with initial conditions matching eigenvectors);
2. write the general solution;
3. determine the particular solutions, each corresponding to the following initial condi-
tions
– x(1) (0) = (1, 0)
– x(2) (0) = (0, 1)
– x(3) (0) = (0, 0)
– x(4) (0) = (−1, 1)
– x(5) (0) = (2, 3)
– x(6) (0) = (1, −3)
– x(7) (0) = (−1, −1)
4. with the aid of the direction field, draw the trajectories originated from each of the
porevious initial conditions;
5. show that limt→+∞ x(t) = 0, for all initial conditions x0 .
• Geven the affine (or non-homogeneous) linear system of the first order ẋ = Ax+b defined
by
ẋ1 −1 1 x1 3
= +
ẋ2 1 −3 x2 −2
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1 0 3
1. verify that the general solution is x(t) = A1 e−t +A2 e−3t +
0 1 −2/3
2. verify that the constant function x(t) = −A−1 b is a particular (fixed point) solution;
3. determine the particular solutions with the following initial conditions
– x(1) (0) = (3, 2)
– x(2) (0) = (−3, −2/3)
– x(3) (0) = (3, −2/3)
– x(4) (0) = (−1, 1)
– x(5) (0) = (2, 3)
– x(6) (0) = (1, −3)
– x(7) (0) = (−1, −1)
4. specify the time dependence of each dynamic variables x 1 and x2 along the trajecto-
ries with the initial conditions above specified; also, provide graphical representations
of x1 (t) and x2 (t);
5. show that the variable z := x + A−1 b satisfy the linear (homogeneous) equation
ż = Az
• Consider a linear system ẋ = Ax, where A is a 2 × 2 matrix. Choose A such that 0 is
the unique fixed point of the system and, alternatively, this fixed point is
1. a stable node;
2. a stable focus:
3. an unstable focus;
4. an unstable node;
5. a saddle.
In each case, sketch the phase space of the system with eigenvalues and lines of null
variations of state variables. Finally, choose A such that 0 is not the unique fixed point.
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• Rewrite the followins ODE as systems of differential equations of first order
1. ẍ + x = 1
...
2. x + 3ẍ − 2x = 0
d4 1
3. 4 x + 4ẍ − ẋ − x = 2
dt 2
ẍ1 + x = 1
4.
ẍ2 + ẋ2 − x2 + x1 = −1
5. ẍ − tx = t2 + 2
• Consider the following linear system ẋ = Ax defined by
ẋ1 1 −2 x1
=
ẋ2 −3 2 x2
find two independent eigenvators v 1 and v 2 of A and verify with a direct computation
that the function
x = 3e4t v 1 − 5e−t v 2
is a solution of the system.
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