10.2 Generalized Eigenvectors
In the previous section we looked at the case where each eigenvalue of a square matrix A
has as many linearly independent eigenvectors as its multiplicity. In that case we could
diagonalize A and use this to compute its power
6.2 Complex Eigenvalues
Let's look an example of a matrix whose eigenvalues are complex numbers.
Example 1. Find the eigenvalues and eigenvectors of A = . For the eigenvalues one has
A - I =
0 = det( A - I ) =
= (- 1 - )(- 3 - ) (1)(- 26)
= 2 + 4 + 3 + 26
6.5 Diagonalization of Matrices with Complex Eigenvalues
In section 6.3 we looked at the diagonalization of a matrix A, i.e.
(1)
A = TDT-1 = TT-1
where 1,m are the eigenvalues of A and v1,vm are the corresponding eigenvectors
and
T = matrix whose columns
9.2 Coordinate Systems and Bases
In this section we want to look at generalized coordinate systems for planes and higher
dimensional subspaces. In a Cartesian coordinate system the coordinate axes are usually
perpendicular. As we have seen earlier it is o
6.4 Difference Equations
In a difference equation the unknowns are one or more sequences of numbers, e.g.
x0, x1, x2, x3, , xn,
y0, y1, y2, y3, , yn,
In applications we are often observing some system that is changing with time and x and
y represent physi
8 Symmetric Matrices and Quadratic Functions
So far most of the applications of matrices have been to linear functions. In this chapter
we look at applications fo matrices to quadratic functions. It turns out that symmetric
matrices play an important role
7.2 Projection on a Plane
In the previous section we looked at projections on a line. In this section we extend this
to projections on planes.
Problem. Given a plane P through the origin and a point v,
find the point w on P closest to v.
v
P
A plane throu
9 Subspaces and Bases
We have seen a number of situations when it is convenient to use a different coordinate
system from the original coordinate system. In this chapter we explore this further and
consider coordinate systems for planes in three dimension
8.2 Orthogonal Matrices
The fact that the eigenvectors of a symmetric matrix A are orthogonal implies the
coordinate system defined by the eigenvectors is an orthogonal coordinate system. In
particular, it makes the diagonalization A = TDT-1 of the matrix
8.3 Quadratic Functions
In this section we shall look at the connection between quadratic functions and symmetric
matrices.
Definition 1. A quadratic function z = f(x, y) of two variables x and y is one of the form
z = f(x, y) = ax2 + 2bxy + cy2 + dx + ey
10 Eigenvalues of Multiplicity Greater Than One
So far we have restricted out attention to matrices whose eigenvalues have multiplicity
one. In this chapter we consider matrices which have eigenvalues of multiplicity greater
than one. Suppose A is a matri
6.3 Diagonalization and Matrix Powers
If A is a square matrix then An = AAA is A multiplied by itself n times.
Example 1. If A = , then
A2 =
=
=
A3 = A2A =
=
=
There is a nice formula for An in terms of the eigenvalues and eigenvectors of A. This
formula