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Math 221 Spring 1998 Computer Assignment 3
due Thursday April 16
- The general equation of a circle is
d(x2+y2)+ax+by+c=0 with .) Given this, show that
the equation of a circle through the points (x1,y1), (x2,y2), and
(x3,y3) is given by
as long as the coefficient of x2+y2 (when this determinant is expanded)
is not zero. HINT: The most direct way to do this problem is to show that
the determinant condition expands to the equation of a circle, and
that the three points actually lie on this circle.
It is also possible
to derive this equation (and many useful others ...)
by writing down a related homogeneous square
system of linear equations with a nontrivial solution, and thinking
about what that says of the determinant of the associated matrix.
But you don't need to do that derivation.
- Using the determinant of part a), find the equation of the circle
through the points (1,2), (3,4), and (5,5). Note that the Maple
may be used to construct a matrix A with rows v1, v2, v3, and v4.
- Try to use this procedure to determine the circle through
(1,2), (3,4), and (5,6). Explain algebraically and geometrically
why any difficulties you run into arise. (The algebraic explanation
should include the vanishing of a certain determinant ....)
- Generalize the method of part a) using determinants to find a conic
ax2+bxy+cy2+dx+ey+f=0 though the points (1,2), (3,4),
(5,6), (7,9), and (9,11).
The method is readily generalized to n x n matrices.
- Suppose that a 2 x 2 matrix A has 2 linearly independent
with respective eigenvalues
Suppose that a vector
can be written as
Give a formula in terms of this data
for the n'th power of A acting on ,
explain why the direction of
will approach that of
the power n gets large.
- The Maple below implements this power method for trying to find a unit eigenvector corresponding to the eigenvalue of largest absolute value.
w := vector([1,0]);
for i from 1 to 10 do
w := evalm(A &* w);
w_len := evalf(sqrt(dotprod(w,w))):
w := map(evalf,evalm(1/w_len * w)):
( You can use the key option-return instead of return to insert a
new line without generating a Maple prompt. )
Using the initial vectors
whether the power method as above seems to be approaching an eigenvector of
so, which eigenvector ? How long does each take to compute a unit eigenvector to two decimal places ?
- Use the initial vector
Explain why your result differs from part c) above.. Will the power method
produce an eigenvector in this case ?
For problem 3 you may wish to use the following Maple commands
(on top of the ones you already know).
B :=submatrix(A, [1,2], [2,3]); (B is the respective sub-matrix of A)
nullspace(A); (calculate the null-space of matrix of A)
rank(A); (calculate the rank of matrix of A)
charpoly(A,lambda); (calculate the characteristic polynomial of matrix of A
in the unknown )
solve(pol=0,x);(find the roots of the polynomial pol with respect to the unknown x)
Consider the matrix
Note: This exercise illustrates that
it is necessary to use a system like Maple in a smart way
in order to get the desired information. In particular, it is important to have an
idea of the underlying theory.
- Find the characteristic polynomial of A and the eigenvalues as its roots.
Clearly the dimension of the eigenspace to the eigenvalue -2 is at least one.
It is the purpose of what follows to find
conditions on a, b and c so that this eigenspace has
dimension two or
- Calculate the rank of A+2I.
Note that Maple takes a,b and c to be arbitrary numbers, so that
it finds that A+2I has rank two, meaning that the eigenspace of -2 is of dimension
This should be interpreted in the following sense: ``The eigenspace has at least
dimension one, and for typical a,b,c it has dimension exactly one''.
In other words, the Maple rank command does not give you the desired
conditions on a, b and c.
- To overcome this problem
consider the top-left
H of A+2I. Find conditions on a,b and c for when the determinant
of H is zero.
(If this determinant is zero the rank of A+2I can be less than two,
meaning that the eigenspace of -2 can have dimension bigger than one.)
- Plug in the conditions on a,b and c you found in c) into the matrix
A+2I one by one. This allows you to determine the
exact conditions on a,b and c so that A+2I has rank one or zero, which
answers the question regarding
the dimension of this eigenspace of -2.
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