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 nn 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 33 case.

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by definition

The three columns are linearly independent becausethe matrix is invertible

Proof continued

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and

and

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 nn matrix M has n distinct eigenvalues,then M is diagonalizable

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The converse is false: There is some diagonalizable matrix with repeated eigenvalues.

Matrix in special form

• Symmetric: AT=A.• Skew-symmetric: AT= –A.• Orthogonal: AT =A-1, or equivalently AT A = I.• Examples:

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symmetric skew-symmetricsymmetricandorthogonal

Orthogonal matrix

A matrix M is called orthogonal if

Each column has norm 1

Dot product = 1

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MMT I

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 nn 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.

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Proof omitted.

Q

http://en.wikipedia.org/wiki/Principal_axis_theorem

Two sufficient conditions for diagonalizability

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Diagonalizable

Distinct eigenvaluesSymmetric,

skew-symmetric,orthogonal

Example

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Similarity• Definition: We say that two nn 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

Change coordinates

Hyperbola

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Diagonalize

x2 – 4xy +2y2 = 1

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

x

y

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2x2 + 2xy + 2y2 = 1Rewrite using symmetric matrix

Compute the characteristic polynomial

Find the eigenvalues

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2x2 + 2xy + 2y2 = 1Diagonalize

Change of variables

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Columns of P are eigenvectors,normalized to norm 1.

2x2 + 2xy + 2y2 = 1

-0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x

y

u

v

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