Additional notes on linear algebra |
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© Leon van Dommelen |
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10 Finding eigenvalues
In this class, to find the eigenvalues of an matrix,
- Form the matrix . That means, add to
each diagonal elements. (Don't forget zero diagonal elements.)
- Find the determinant of that matrix using the method of
minors. (Gaussian elimination is too messy here and should not
be used.)
- Set this determinant to zero. For an matrix,
the determinant can always be written in the form
where are the
eigenvalues. They are found as the roots (or zeros) of the
determinant. There are always eigenvalues. But these
eigenvalues do not necessarily correspond to different numbers.
For example, for some matrix might be the same number as
. In that case, that number is called a “double
eigenvalue”. If three of the eigenvalues are the same
number, that number is called a triple eigenvalue,
etcetera.
For an matrix, the determinant is always a polynomial of
degree , call it . Now finding the roots of
quadratics, , is easy. But if the dimension of the matrix ,
you have to solve a cubic equation. For that the general solution is
very and hard to apply, especially if you do not know complex
variables. For , the general solution is even worse, and for
or more, there is no general expression for the roots at all.
(It has in fact been proved that such an expression is impossible to
find.) To deal with such problems, here are some tricks:
- Do not be too quick to multiply out. Maybe you can recognize a
common factor in all terms before multiplying
out. In that case, is one of your eigenvalues. To find
the remaining eigenvalues, take the factor out
of the entire expression and look at what is left.
- If you can guess a root (by trying, say, 0, ,
, ..., and seeing whether the determinant is zero for that
value of ), then you can write the original characteristic
polynomial as
where is one degree less that . So
its roots are easier to find. You find by long
division of by .
- Sometimes a fourth order polynomial is really just a quadratic
when written in terms of , like
. In that case, first find the values
of , then from those the values of .