MATH2019 PROBLEM 6- MATRICES Solution

EXAMPLES6
MATRICES
1991
&
1994
1. a) Find the eigenvalues and the corresponding eigenvectors of matrix
A .
b) Find an orthogonal matrix P such that
D = P−1AP
is a diagonal matrix and write down the matrix D.
c) Using your results from parts a) and b) find the solution of the system of differential
equations
dx

dt = 3x + 2y + 2z ,
dy

dt = 2x + 2y ,
dz

dt
subject to the conditions = 2x + 4z ,
x(0) = 0,
d) Express the quadric surface y(0) = 0 and z(0) = 1.
3x2 + 2y2 + 4z2 + 4xy + 4xz = 24
in terms of its principal axes X, Y and Z and write out an orthogonal matrix P such that
.
What shape does this quadric surface represent?
e) A and P are n×n matrices. A is symmetric and P is orthogonal. Prove that P−1AP is symmetric.
1998 2. Let
A
a) Find the eigenvalues and eigenvectors of A.
b) Normalise the eigenvectors to have length 1. Hence find an orthogonal matrix P such that
D = P−1AP
is a diagonal matrix. Evaluate both sides of this equation to show that it is satisfied by your P.
c) For the system of differential equations
dx
= Ax where x dt
show (or verify) that the transformation
x = Pz where z
yields the equation
dz
= Dz dt
where P and D are as in part b).
d) Hence solve the system of equations
dx
= Ax dt
if x1(0) = 1, x2(0) = 0.
1999 3. Let A .
a) Find the eigenvalues and eigenvectors of A.
b) i) Find a matrix P such that P−1AP = D where D is a diagonal matrix.
ii) Calculate P−1AP to check this is indeed equal to D.
c) If x and f show that with the definition x = Pz, the system of differential equations
dx
= Ax + f (1)
dt
becomes
dz −1
= Dz + P f .
dt
d) Using the result of c) find the general solution of (1) in the case when f1(t) = e2t and f2(t) = 0.
2000 4. Consider the quadric surface given by
x2 + y2 + 3z2 + 4xz + 4yz = 5.
a) Express this equation in the form
vTAv = 5
where A is a real symmetric matrix and v .
b) Show that the matrix A has an eigenvalue λ = 1 and two other distinct eigenvalues.
What are the values of these other eigenvalues?
c) Write down the equation of the quadric surface in terms of its principal axes X, Y and Z. Then sketch the surface relative to principal axes, clearly labelling the (X,Y,Z) coordinates of the points where the surface intersects the principal axes.
d) Find the eigenvectors of A and hence find an orthogonal matrix, P, which relates  X 
and Y . Write down this relationship.  Z 
e) Write down the points of intersection of the quadric surface with its principal axes in terms of the (x,y,z) coordinate system.
2014, S1 5. It is given that the matrix A has an eigenvalue λ1 = 1
with an associated eigenvector v and eigenvalue λ2 = 6
with associated eigenvector v .
a) Without calculating the characteristic polynomial explain why the remaining eigen-value is λ3 = 7.
b) Find an eigenvector vλ=7 for the eigenvalue λ3 = 7.
c) Hence write down the general solution to the system of differential equations
y1′ = y1 y2′ = −8y1 + 4y2 − 6y3 y3′ = 8y1 + y2 + 9y3
2014, S2 6. Let
A .
a) Find the eigenvalues and eigenvectors of the matrix A.
b) By considering the eigenvalues of A, write the curve
2x2 + 6xy + 2y2 = 45
in terms of principle axes coordinates X and Y . Sketch the curve in the XY -plane.
c) Find the distance from the curve 2x2 + 6xy + 2y2 = 45 to the origin.
2015, S1 7. The equations governing the response of a bridge to an earthquake are found to satisfy
dx
= −x + ay,
dt
dy
= ax − y. dt
where a > 0 is a parameter that depends on the material used for the bridge.
a) Express this set of differential equations in the form
dx x
= Ax, where x . dt
and find the eigenvalues and eigenvectors of the matrix A.
b) Hence, or otherwise, write down a general solution for the problem.
c) For what values of a will the solution grow with increasing t?
2015, S2 8. The matrix B is given by
B .
a) Show that the vector
v
is an eigenvector of the matrix B and find the corresponding eigenvalue.
x2 − 6xy + y2 = 16.
a) Rewrite the equation for the curve in the form

where A is a real symmetric 2 × 2 matrix. Find the eigenvalues and eigenvectors of A.
b) Write down the equation for the curve in terms of its principle axes X and Y . Hence find the closest distance from the origin to the curve.
c) Find the x and y coordinates of the points on the curve closest to the origin.
a) Find the eigenvalues and eigenvectors of A.
b) Hence solve the system of differential equations
dx
= 6x + 2y
dt
dy
= −x + 3y. dt
dx
= −x + y,
dt
dy
= x − y, dt
with initial conditions x(0) = 1, y(0) = 0.
a) Express this set of differential equations in the form
dx x
= Ax, where x . dt
and find the eigenvalues and eigenvectors of the matrix A.
b) Hence, or otherwise, write down the solution for the problem using the initial con-ditions.
a) Express the curve in the form
xTAx = 1, where x ,
and find the eigenvalues and eigenvectors of the matrix A.
b) Hence, or otherwise, find the distance from the curve to the origin. Write down thex and y coordinates of the points on the curve closest to the origin.
xTAx = 12 where x .
A student is given the following extra information about matrix A:
– trace(A) = 0,
– λ1 = 2 and λ3 = 4 with associated eigenvectors, respectively,
v and v .
a) What is the value of the remaining eigenvalue, namely λ2?
b) Write down the equation of the quadric surface, relative to the principal axes of thesurface.
c) Write down a vector v2 that is orthogonal to both eigenvectors v1 and v3.
d) What is the relationship between λ2 and v2? Give reasons for your answer.
e) Hence determine an orthogonal matrix P which diagonalises the matrix A such that P−1AP = D where D is a 3 × 3 diagonal matrix.
f) Hence determine the matrix A.
i) Express the curve in the formxTAx = 200
where x and A is a 2 × 2 real symmetric matrix.
ii) Find the eigenvalues and eigenvectors of the matrix A in part i). iii) Hence, or otherwise, find the shortest distance between the curve and the origin.