Eigencircles of 2 2 matrices Graham Farr Faculty of IT Monash - PowerPoint PPT Presentation

Eigencircles of 2 × 2 matrices Graham Farr Faculty of IT Monash University Graham.Farr@infotech.monash.edu.au 18 July 2007 Joint work with Michael Englefield (School of Mathematical Sciences, Monash)

Eigenvalues and eigenpairs Eigenvalue of a 2 × 2 matrix: number λ such that � a � � x � x � � b = λ c d y y with x , y not both 0. To start with, λ ∈ R .

Eigenvalues and eigenpairs Eigenvalue of a 2 × 2 matrix: number λ such that � a � � x � x � � b = λ c d y y with x , y not both 0. To start with, λ ∈ R . � λ � 0 Field isomorphism: λ ← → 0 λ

Eigenvalues and eigenpairs Eigenvalue of a 2 × 2 matrix: number λ such that � a � � x � λ � � x � � b 0 = c d y 0 λ y with x , y not both 0. To start with, λ ∈ R . � λ � 0 Field isomorphism: λ ← → 0 λ

Eigenvalues and eigenpairs Eigenvalue of a 2 × 2 matrix: number λ such that � a � � x � λ � � x � � b 0 = c d y 0 λ y with x , y not both 0. To start with, λ ∈ R . � λ � 0 Field isomorphism: λ ← → 0 λ � � λ µ Extend using field isomorphism: λ + µ i ← → − µ λ

Eigenvalues and eigenpairs Eigenvalue of a 2 × 2 matrix: number λ such that � a � � x � λ � � x � � b 0 = c d y 0 λ y with x , y not both 0. To start with, λ ∈ R . � λ � 0 Field isomorphism: λ ← → 0 λ � � λ µ Extend using field isomorphism: λ + µ i ← → − µ λ Eigen pair of a 2 × 2 matrix: ( λ, µ ) ∈ R 2 such that � a � � x � � x � � � b λ µ = c d y − µ λ y with x , y not both 0.

The Eigencircle � a � � x � � x � � � b λ µ = c d y y − µ λ

The Eigencircle � a � � x � � x � � � b λ µ = c d y y − µ λ Eigenpairs must satisfy � � a − λ b − µ � � � = 0 � � c + µ d − λ �

The Eigencircle � a � � x � � x � � � b λ µ = c d y y − µ λ Eigenpairs must satisfy � � a − λ b − µ � � � = 0 � � c + µ d − λ � Some eigenpairs: ( a , b ) , ( a , − c ) , ( d , b ) , ( d , − c ).

The Eigencircle � a � � x � � x � � � b λ µ = c d y y − µ λ Eigenpairs must satisfy � � a − λ b − µ � � � = 0 � � c + µ d − λ � Some eigenpairs: ( a , b ) , ( a , − c ) , ( d , b ) , ( d , − c ). Eigenpairs form a circle, the eigencircle : � 2 � 2 � 2 � 2 � λ − a + d � � a + d � b − c µ − b − c + = + − ( ad − bc ) 2 2 2 2 ) 2 + ( µ − ) 2 f 2 g 2 ( λ − f g = + det A −

The Eigencircle µ − c b a O d λ

The Eigencircle µ H E − c b F G a O d λ

The Eigencircle µ H E − c C b F G a O d λ

The Eigencircle µ H E − c g C b F G a O f d λ

The Eigencircle µ − c g C b a O f d λ

The Eigencircle µ − c g C b R a O f d λ

The Eigencircle µ − c C b R a O d λ

The Eigencircle µ − c C ρ b R √ det A a O d λ

The Eigencircle µ − c C ρ b R √ . . . provided det A > 0 det A a O d λ

The Eigencircle: det A < 0

The Eigencircle: det A < 0 µ C O λ

The Eigencircle: det A < 0 µ C R O λ

The Eigencircle: det A < 0 µ C R ρ O λ √ − det A

The Eigencircle: det A = 0

The Eigencircle: det A = 0 µ C O λ

The Eigencircle: det A = 0 µ C R = ρ O λ

The Eigencircle Determinant    outside > 0    Origin is on  eigencircle det A = 0 ⇐ ⇒ inside < 0  

The Eigencircle Determinant    outside > 0    Origin is on  eigencircle det A = 0 ⇐ ⇒ inside < 0   Real eigenvalues Eigencircle meets λ -axis ⇐ ⇒ eigenvalues are real

Eigenvectors

Eigenvectors Given (real) eigenvalue λ , � a � � x � λ � � x � � b 0 = 0 c d y λ y � x � λ � � � � d get eigenvectors: = any multiple of . − 0 y − c

Eigenvectors Given (real) eigenvalue λ , � a � � x � λ � � x � � b 0 = 0 c d y λ y � x � λ � � � � d get eigenvectors: = any multiple of . − 0 y − c µ ( d , − c ) C O ( λ 1 , 0) ( λ 2 , 0) λ

Eigenvectors For a real symmetric 2 × 2 matrix, distinct real eigenvalues have perpendicular eigenvectors.

Eigenvectors For a real symmetric 2 × 2 matrix, distinct real eigenvalues have perpendicular eigenvectors. Proof without words: µ ( d , − c ) C O ( λ 1 , 0) ( λ 2 , 0) λ

( λ, µ )-eigenvectors „ x « A ( λ, µ ) -eigenvector is a nonzero corresponding to the y eigenpair ( λ, µ ).

( λ, µ )-eigenvectors „ x « A ( λ, µ ) -eigenvector is a nonzero corresponding to the y eigenpair ( λ, µ ). µ ( d , − c ) ( λ, µ ) C O λ

( λ, µ )-eigenvectors „ x « A ( λ, µ ) -eigenvector is a nonzero corresponding to the y eigenpair ( λ, µ ). µ ( d , − c ) ( λ, µ ) C O λ Diametrically opposite eigenpairs have perpendicular ( λ, µ ) -eigenvectors

( λ, µ )-eigenvectors „ x « A ( λ, µ ) -eigenvector is a nonzero corresponding to the y eigenpair ( λ, µ ). µ ( d , − c ) ( λ, µ ) C O λ ( λ ′ , − µ ′ ) Diametrically opposite eigenpairs have perpendicular ( λ, µ ) -eigenvectors

Power and determinant R C Q Euclid’s Elements III.35–36: Power of point = PQ · PR , independent of direction of line P

Power and determinant R C Q Euclid’s Elements III.35–36: Power of point = PQ · PR , independent of direction of line P

Power and determinant C R Q Euclid’s Elements III.35–36: Power of point = PQ · PR , independent of direction of line P

Power and determinant C R Q Euclid’s Elements III.35–36: Power of point = PQ · PR , independent of direction of line P

Power and determinant C Q = R Euclid’s Elements III.35–36: Power of point = PQ · PR , independent of direction of line P

Power and determinant eigencircle C Q = R Euclid’s Elements III.35–36: Power of point = PQ · PR , independent of direction of line P

Power and determinant eigencircle C µ Q = R Euclid’s Elements III.35–36: Power of point = PQ · PR , independent of direction of line O = P λ

Power and determinant eigencircle C µ Q = R √ Euclid’s Elements III.35–36: det A Power of point = PQ · PR , independent of direction of line O = P λ

Power and determinant eigencircle det A = power of origin w.r.t. eigencircle C µ Q = R √ Euclid’s Elements III.35–36: det A Power of point = PQ · PR , independent of direction of line O = P λ

Power and determinant

Power and determinant R C P Q Power of point = PQ · PR ; lengths are signed , so now < 0

Power and determinant R C P Q Power of point = PQ · PR ; lengths are signed , so now < 0

Power and determinant C R P Power of point = PQ · PR ; Q lengths are signed , so now < 0

Power and determinant C R P Power of point = PQ · PR ; Q lengths are signed , so now < 0

Power and determinant eigencircle C R P Power of point = PQ · PR ; Q lengths are signed , so now < 0

Power and determinant µ eigencircle C R O λ Power of point = PQ · PR ; Q lengths are signed , so now < 0

Power and determinant µ eigencircle C R O λ Power of point = PQ · PR ; Q lengths are signed , so now < 0 √ − det A

Power and determinant µ det A = power of origin eigencircle w.r.t. eigencircle C R O λ Power of point = PQ · PR ; Q lengths are signed , so now < 0 √ − det A

Power and discriminant

Power and discriminant λ 2 − ( a + d ) λ + ( ad − bc ) = 0 Characteristic equation of A :

Power and discriminant λ 2 − ( a + d ) λ + ( ad − bc ) = 0 Characteristic equation of A : Real eigenvalues:

Power and discriminant λ 2 − ( a + d ) λ + ( ad − bc ) = 0 Characteristic equation of A : Real eigenvalues: Discriminant: ( a + d ) 2 − 4 det A ∆ = µ 4( f 2 − det A ) = N 4( ρ 2 − g 2 ) = = − 4( g − ρ )( g + ρ ) = − 4 · YM · YN = − 4 · (power of Y ) C Y O λ L 1 L 2 ( λ 1 , 0) ( λ 2 , 0) M

Power and discriminant λ 2 − ( a + d ) λ + ( ad − bc ) = 0 Characteristic equation of A : Real eigenvalues: Discriminant: ( a + d ) 2 − 4 det A ∆ = µ 4( f 2 − det A ) = N 4( ρ 2 − g 2 ) = = − 4( g − ρ )( g + ρ ) = − 4 · YM · YN = − 4 · (power of Y ) 4 · ( YL i ) 2 ; C = Y O λ L 1 L 2 ( λ 1 , 0) ( λ 2 , 0) M