Copying is never allowed, even when working together.
An anti-symmetric matrix is a matrix for which .
Are the eigenvalues of an antisymmetric real matrix real too? To
check, write down the simplest nontrivial anti-symmetric
matrix you can think of (which may not be symmetric) and see. In
fact, the eigenvalues of an antisymmetric matrix are always purely
imaginary, i.e. proportional to . The (complex)
eigenvectors are orthogonal, as long as you remember that in the
first vector of a dot product, you must take complex conjugate,
i.e. replace every by . Verify this for your antisymmetric
matrix.
Analyze and accurately draw the quadratic curve
using matrix diagonalization. Show the exact, as well as the
approximate values for all angles. Repeat for the curve,
Note: if you add say 3 times the unit matrix to a matrix , then
the eigenvectors of do not change. It only causes the
eigenvalues to increase by 3, as you can readily verify from the
definition of eigenvector. Use this to your advantage.
Given
Without doing any mathematics, what can you say immediately about
the eigenvalues and eigenvectors of this matrix? Now find the
equation for the eigenvalues. It is a cubic one. However, one
eigenvalue is immediately obvious from looking at . What
eigenvalue , that makes singular, is
immediately obvious from looking at A? Explain. Factor out the
corresponding factor from the cubic, then find
the roots of the remaining quadratic. Number the single eigenvalue
, and the double one and . The
found two basis vectors of the null space of , call
them and , will not be orthogonal
to each other. To make them orthogonal, you must eliminate the
component that has in the direction of . In particular, if is the unit vector
in the direction of , then is the scalar component of in the direction of
. Multiply by the unit vector to get the
vector component, and substract it from :
(This trick of making vectors orthogonal by substracting away the
components in the wrong directions is called
Gram-Schmidt orthogonalization.) Now make this
vector of length 1. Then describe the transformation of basis that
turns matrix into a diagonal one. What is the transformation
matrix from old to new and what is its inverse? What is the
diagonal matrix ? Do your eigenvectors form a right or
left-handed coordinate system?
Consider the quadratic surface given by
Write this in vector-matrix notation and identiy the matrix .
Compare with the matrix of the previous question. Now determine the
shape of the quadratic surface. If it is a spheroid, state whether
it is oblate or prolate. If it is a hyperboloid, state whether it
is one of revolution or not, and whether it is of one or two sheets.
Repeat for the surface
(Note that the new matrix is as if you subtracted 2 times the unit
matrix from the previous matrix. As already noted in an earlier
homework, this does nothing to the eigenvectors, but subtracts 2
from each eigenvalue.)