Transcript The Characteristic Equation of a Square Matrix (11/19/04)

The Characteristic Equation of
a Square Matrix (11/18/05)
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To this point we know how to check
whether a given number  is an
eigenvalue of a given square matrix A ,
but we do not know how to find the
eigenvalues of A (unless A is upper
triangular).
The trick is to use the determinant of A !
Definition
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The characteristic equation of an
n by n square matrix A is the equation
det(A -  I n) = 0
This is a polynomial equation of degree
n in the variable  .
Punch line: The roots of this equation
are the eigenvalues of A .
Two examples
 1 1
= 
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2
4
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Find the eigenvalues of A
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Find the eigenvalues of B = .95 .03
.05 .97
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Why it works
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 is an eigenvalue of a matrix A if and
only if
A x =  x for some nonzero vector x,
which is true if and only if
(A -  I ) x = 0 has a non-trivial
solution, which is true if and only if
det(A -  I ) = 0 .
Is that cool or what?
But there are drawbacks…
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First, determinants are not easy to
compute, as we well know.
Second, once the determinant is found,
the polynomial equation may not be
easy to solve.
For example, what are the eigenvalues
of matrices whose characteristic
equations are:
1. 2 - 3 + 4 = 0
2. 3 - 2 2 + 5  - 3 = 0
Assignment for Monday
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Finish up HW #3.
Read Section 5.2. (Note: We already
know the material on the determinant.
They are assuming that the reader
skipped Chapter 3)
Do Exercises 1-11 odd, 15, and 19.