ENGG2013 Unit 19
The principal axes theorem
Mar, 2011.
Outline
• Special matrices
– Symmetric, skew-symmetric, orthogonal
• Principle axes theorem
• Application to conic sections
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Diagonalizable ??
• 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.
– Example
• Some matrix cannot be diagonalized.
– Example
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Theorem
An nn matrix M is diagonalizable if and only if we can
find n linear independent eigenvectors of M.
Proof: For concreteness, let’s just consider the 33 case.
The three
columns are
linearly
independent
because
the matrix is
invertible
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by definition
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Proof continued
and
and
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Complex eigenvalue
• There are some matrices whose eigenvalues
are complex numbers.
– For example: the matrix which represents rotation
by 45 degree counter-clockwise.
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Theorem
If an nn matrix M has n distinct eigenvalues,
then M is diagonalizable
The converse is false:
There is some diagonalizable matrix with repeated eigenvalues.
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Matrix in special form
•
•
•
•
Symmetric: AT=A.
Skew-symmetric: AT= –A.
Orthogonal: AT =A-1, or equivalently AT A = I.
Examples:
symmetric
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skew-symmetric
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symmetric
and
orthogonal
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Orthogonal matrix
A matrix M is called orthogonal if
Each column has norm 1
MT
M
I
Dot product = 1
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Orthogonal matrix
A matrix M is called orthogonal if
Any two distinct columns are orthogonal
Dot product = 0
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Principal axes theorem
Given any nn symmetric matrix A, we have:
1. The eigenvalues of A are real.
2. A is diagonalizable.
3. We can pick n mutually perpendicular (aka
orthogonal) eigenvectors.
Q
Proof omitted.
http://en.wikipedia.org/wiki/Principal_axis_theorem
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Two sufficient conditions for
diagonalizability
Symmetric,
skew-symmetric,
orthogonal
Distinct eigenvalues
Diagonalizable
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Example
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Similarity
• Definition: We say that two nn matrix A and B
are similar if we can find an invertible matrix S
such that
• Example:
and
are similar,
• The notion of diagonalization can be phrased in
terms of similarity: matrix A is diagonalizable if
and only if A is similar to a diagonal matrix.
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More examples
•
is similar to
because
•
and
are similar.
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Application to conic sections
• Ellipse : x2/a + y2/b = 1.
• Hyperbola : x2/a – y2/b = 1.
• Parabola y = ax2.
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Application to conic sections
• Is x2 – 4xy +2y2 = 1 a ellipse, or a hyperbola?
Rewrite using symmetric matrix
Find the characteristic polynomial
Solve for the eigenvalues
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Application to conic sections
Diagonalize
Change coordinates
Hyperbola
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x2 – 4xy +2y2 = 1
15
10
y
5
0
-5
-10
-15
-15
-10
-5
0
5
10
15
x
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2x2 + 2xy + 2y2 = 1
Rewrite using symmetric matrix
Compute the characteristic polynomial
Find the eigenvalues
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2x2 + 2xy + 2y2 = 1
Columns of P are eigenvectors,
normalized to norm 1.
Diagonalize
Change of variables
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2x2 + 2xy + 2y2 = 1
v
0.8
0.6
0.4
y
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
0.5
1
u
x
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