Download ENGG2013 Lecture 17

ENGG2013 Unit 17
Diagonalization
Eigenvector and eigenvalue
Mar, 2011.
EXAMPLE 1
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Q6 in midterm
• u(t): unemployment rate in the t-th month.
• e(t)= 1-u(t)
• The unemployment rate in the next month is
given by a matrix multiplication
• Equilibrium: Solve
 Unemployment rate at equilibrium = 0.2
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Equilibrium
Unstable
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Stable
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If stable, how fast does it converge
to the equilibrium point?
Slow convergence
0.2
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Fast convergence
0.2
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Question
• Suppose that the initial unemployment rate at
the first month is x(1), (for example x(1)=0.25),
and suppose that the unemployment evolves
by matrix multiplication
Find an analytic expression for x(t), for all t.
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EXAMPLE 2
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How to count?
• Count the number of binary strings of length n
with no consecutive ones.
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SOLVING RECURRENCE RELATION
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•
•
•
•
F1 = 1
F2 = 1
For n > 2, Fn = Fn-1+Fn-2.
The Fibonacci numbers are
– 1,1,3,5,8,13,21,34,55,89,144
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http://en.wikipedia.org/wiki/Fibonacci_number
Fibonacci numbers
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A matrix formulation
• Define a vector
• Initial vector
• Find the recurrence relation in matrix form
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A general question
• Given initial condition
and for t  2
Find v(t) for all t.
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Matrix power
• Need to raise a matrix to a very high power
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A trivial special case
• Diagonal matrix
• The solution is easy to find
• Raising a diagonal matrix to the power t is
easy.
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Decoupled equations
• When the equation is diagonal, we have two
separate equation, each in one variable
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DIAGONALIZATION
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Problem reduction
• A square matrix M is called diagonalizable if
we can find an invertible matrix, say P, such
that the product P–1 M P is a diagonal matrix.
• A diagonalizable matrix can be raised to a high
power easily.
– Suppose that P–1 M P = D, D diagonal.
– M = P D P–1.
– Mn = (P D P–1) (P D P–1) (P D P–1) … (P D P–1)
= P Dn P–1.
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Example of diagonalizable matrix
• Let
• A is diagonalizable because we can find a
matrix
such that
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Now we know how fast it
converges to 0.2
• The matrix can be diagonalized
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Convergence to equilibrium
• The trajectory of the unemployment rate
– the initial point is set to 0.1
0.2
0.19
0.18
Unemployment rate
0.17
0.16
0.15
0.14
0.13
0.12
0.11
0.1
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month (t)
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EIGENVECTOR AND EIGENVALUE
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How to diagonalize?
• How to determine whether a matrix M is
diagonalizable?
• How to find a matrix P which diagonalizes a
matrix M?
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From diagonalization to
eigenvector
• By definition a matrix M is diagonalizable if
P–1 M P = D
for some invertible matrix P, and diagonal
matrix D.
or equivalently,
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The columns of P are special
• Suppose that
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Definition
• Given a square matrix A, a non-zero vector v is
called an eigenvector of A, if we an find a real
number  (which may be zero), such that
Matrix-vector product
Scalar product of a vector
• This number  is called an eigenvalue of A,
corresponding to the eigenvector v.
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Important notes
• If v is an eigenvector of A with eigenvalue ,
then any non-zero scalar multiple of v also
satisfies the definition of eigenvector.
k0
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Geometric meaning
• A linear transformation L(x,y) given by: L(x,y) = (x+2y, 3x-4y)
x  x + 2y
y  3x – 4y
• If the input is x=1, y=2 for example,
the output is x = 5, y = -5.
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Invariant direction
• An Eigenvector points at a direction which is invariant under the linear
transformation induced by the matrix.
• The eigenvalue is interpreted as the magnification factor.
• L(x,y) = (x+2y, 3x-4y)
• If input is (2,1), output is magnified by a factor of 2, i.e., the eigenvalue is 2.
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Another invariant direction
•
•
L(x,y) = (x+2y, 3x-4y)
If input is (-1/3,1), output is (5/3,-5). The length is increased by a factor of 5, and
the direction is reversed. The corresponding eigenvalue is -5.
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Eigenvalue and eigenvector of
First eigenvalue = 2, with eigenvector
where k is any nonzero real number.
Second eigenvalue = -5, with eigenvector
where k is any nonzero real number.
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Summary
• Motivation: want to solve recurrence
relations.
• Formulation using matrix multiplication
• Need to raise a matrix to an arbitrary power
• Raising a matrix to some power can be easily
done if the matrix is diagonalizable.
• Diagonalization can be done by eigenvalue
and eigenvector.
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