Let
for
and
Suppose that
be two bases
Which of the following is true?
1.
2.
3.
4.
(a)
(b)
(c)
(d)
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Definition. An eigenvector of a square matrix
is a
nonzero vector
such that
for some
scalar
A scalar
is called an eigenvalue of
if there is
a nontrivial solution
of
such an
is
called an eigenvector of
corresponding to
The set of all solutions to
called the eigenspace of
eigenvalue
is
corresponding to the
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To find the eigenvalues of
Characteristic polynomial:
Characteristic equation:
3
Example. Let
eigenvalues of
Find all the
and give a basis for each eigenspace.
4
Question. What are the eigenvalues of a triangular
matrix?
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Example. Find all the eigenvalues of
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Application to Dynamical Systems
(see also: sections 1.10, 4.9)
Suppose demographic studies show that each year about 5%
of a city’s population moves to the suburbs (and 95%
remains in the city), while 10% of the suburban population
moves in the city (and 90% remains in the suburbs).
Let
initial city population
initial suburbs population.
Then after one year,
This is equivalent to
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Let
Then
Let
(population vector after
years).
Then the dynamical system is given by
Assume that
(in ten thousands).
Analyze the long-term behavior of this dynamical system.
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Procedure:
(1) Find the eigenvalues of M and a basis for each
eigenspace.
(2) Write
in terms of the basis vectors obtained in (1).
(3) Find a formula for
in terms of the eigenvectors
corresponding to the eigenvalues by using linearity.
(4) Let
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Theorem. A is invertible if and only if _______ is
not an eigenvalue of A.
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Let
Is
an eigenvector of
1. Yes
2. No
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Similarity
For square matrices and
we say that
similar to
if there is an invertible matrix
such that
is
Theorem. Similar matrices have the same
determinant.
Proof.
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5.3 Diagonalization
The goal here is to develop a useful factorization
where
is diagonal. We can
use this to compute
quickly when
is large.
If
then
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Example. Let
In general, what is
integer?
Compute
where
is any positive
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Example. Let
It can be shown that
where
Find
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Definition. A square matrix
is said to be
diagonalizable if
is similar to a diagonal matrix,
i.e. if
where
is invertible and
is a diagonal matrix.
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When is A diagonalizable?
17
Theorem.
is diagonalizable if and only if
linearly independent eigenvectors.
has
18
Example.
eigenvalues:
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Example. Diagonalize the following matrix, if possible:
Step 1. Find the eigenvalues of A.
20
Step 2. Find linearly independent eigenvectors of A.
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Step 3. Construct P from the vectors in Step 2.
Step 4. Construct D from the corresponding eigenvalues.
Step 5. Check that AP = PD.
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Example. Diagonalize the following matrix, if possible.
23
Example. Why is
diagonalizable?
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Theorem. An n x n matrix with n distinct eigenvalues
is diagonalizable.
Warning: The converse is not true!
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