R7003E - Automatic ControlLesson 7

Damiano Varagnolo

14 November 2017

1

Labs

2

Reviews from scalable learning

3

Recap on convolution

4

Recap on linearization

5

Peer instructions

6

Exercises

7

Convolution

y(t) + y(t) = u(t) u(t) = e−2t y(0) = 0

8

Algebraic and geometric multiplicities

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 11

12 1

2 12

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

9

Change of bases

V = [2 11 2] W = [3 1

1 3] x = V [1

0] x =W [x

′1

x′2]

find x′1 and x′2

10

(A bit of) linear algebra

11

Linear transformations and matrices

(By TreyGreer62 - Image:Mona Lisa-restored.jpg, CC0, https://commons.wikimedia.org/w/index.php?curid=12768508)

linear transformation A ≠ matrix A

eigenvectors and eigenvalues are of transformation A, not just of matrix A

12

Linear transformations and matrices

(By TreyGreer62 - Image:Mona Lisa-restored.jpg, CC0, https://commons.wikimedia.org/w/index.php?curid=12768508)

linear transformation A ≠ matrix A

eigenvectors and eigenvalues are of transformation A, not just of matrix A

12

Linear transformations and matrices

(By TreyGreer62 - Image:Mona Lisa-restored.jpg, CC0, https://commons.wikimedia.org/w/index.php?curid=12768508)

linear transformation A ≠ matrix A

eigenvectors and eigenvalues are of transformation A, not just of matrix A

12

Linear transformations and matrices

A ∶D ↦ C D = Rm C = Rn vD1 , . . . , vD

m; vC1 , . . . , vC

n bases of D and C

[AvD1 . . . AvD

m] = [vC1 vC

2 ⋯ vCn ]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ am1a12 ⋯ am2⋮ ⋮

a1n ⋯ amn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ax = [vC1 vC

2 ⋯ vCn ]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ am1a12 ⋯ am2⋮ ⋮

a1n ⋯ amn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

λD1

λD2⋮

λDm

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

13

Linear transformations and square matrices

A ∶ Rn ↦ Rn Ô⇒ C =D

we may choose {vD1 , . . . , vD

n } = {vC1 , . . . , vC

n } = {v1, . . . , vn}

A “ + ” {v1, . . . , vn} ↦ A

A “ + ” {w1, . . . , wn} ↦ A′

how do A and A′ relate?

14

Linear transformations and square matrices

A ∶ Rn ↦ Rn Ô⇒ C =D

we may choose {vD1 , . . . , vD

n } = {vC1 , . . . , vC

n } = {v1, . . . , vn}

A “ + ” {v1, . . . , vn} ↦ A

A “ + ” {w1, . . . , wn} ↦ A′

how do A and A′ relate?

14

Linear transformations and square matrices

A ∶ Rn ↦ Rn Ô⇒ C =D

we may choose {vD1 , . . . , vD

n } = {vC1 , . . . , vC

n } = {v1, . . . , vn}

A “ + ” {v1, . . . , vn} ↦ A

A “ + ” {w1, . . . , wn} ↦ A′

how do A and A′ relate?

14

Linear transformations and square matrices

A ∶ Rn ↦ Rn Ô⇒ C =D

we may choose {vD1 , . . . , vD

n } = {vC1 , . . . , vC

n } = {v1, . . . , vn}

A “ + ” {v1, . . . , vn} ↦ A

A “ + ” {w1, . . . , wn} ↦ A′

how do A and A′ relate?

14

Linear transformations and square matrices

A ∶ Rn ↦ Rn Ô⇒ C =D

we may choose {vD1 , . . . , vD

n } = {vC1 , . . . , vC

n } = {v1, . . . , vn}

A “ + ” {v1, . . . , vn} ↦ A

A “ + ” {w1, . . . , wn} ↦ A′

how do A and A′ relate?

14

Linear transformations and square matrices

A ∶ Rn ↦ Rn Ô⇒ C =D

we may choose {vD1 , . . . , vD

n } = {vC1 , . . . , vC

n } = {v1, . . . , vn}

A “ + ” {v1, . . . , vn} ↦ A

A “ + ” {w1, . . . , wn} ↦ A′

how do A and A′ relate?

14

Changes of bases (summary)

v1, . . . , vn and w1, . . . , wn bases of Rn Ô⇒

Rn ∋ x = [v1 v2 ⋯ vn]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ1λ2⋮

λn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

= [w1 w2 ⋯wn]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

γ1γ2⋮

γn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

[v1 v2 ⋯ vn] = [w1 w2 ⋯wn]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

γ1→1 γ2→1 ⋯ γn→1γ1→2 γ2→2 ⋯ γn→2⋮ ⋮ ⋮

γ1→n γ2→n ⋯ γn→n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

or, compactly, V =WΓv→w

15

Changes of bases (summary)

v1, . . . , vn and w1, . . . , wn bases of Rn Ô⇒

Rn ∋ x = [v1 v2 ⋯ vn]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ1λ2⋮

λn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

= [w1 w2 ⋯wn]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

γ1γ2⋮

γn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

[v1 v2 ⋯ vn] = [w1 w2 ⋯wn]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

γ1→1 γ2→1 ⋯ γn→1γ1→2 γ2→2 ⋯ γn→2⋮ ⋮ ⋮

γ1→n γ2→n ⋯ γn→n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

or, compactly, V =WΓv→w

15

Effects of changing bases on the representations of A

[AvD1 . . . AvD

n ] = [vC1 ⋯ vC

n ]

⎡⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ an1⋮ ⋮

a1n ⋯ ann

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ [Av1 . . . Avn] = [v1 ⋯ vn]

⎡⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ an1⋮ ⋮

a1n ⋯ ann

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ A (V ) = V A

Ô⇒ [Aw1 . . . Awn] = [w1 ⋯wn]

⎡⎢⎢⎢⎢⎢⎢⎣

a′11 ⋯ a′n1⋮ ⋮

a′1n ⋯ a′nn

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ A (W ) =WA′

16

Effects of changing bases on the representations of A

[AvD1 . . . AvD

n ] = [vC1 ⋯ vC

n ]

⎡⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ an1⋮ ⋮

a1n ⋯ ann

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ [Av1 . . . Avn] = [v1 ⋯ vn]

⎡⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ an1⋮ ⋮

a1n ⋯ ann

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ A (V ) = V A

Ô⇒ [Aw1 . . . Awn] = [w1 ⋯wn]

⎡⎢⎢⎢⎢⎢⎢⎣

a′11 ⋯ a′n1⋮ ⋮

a′1n ⋯ a′nn

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ A (W ) =WA′

16

Effects of changing bases on the representations of A

[AvD1 . . . AvD

n ] = [vC1 ⋯ vC

n ]

⎡⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ an1⋮ ⋮

a1n ⋯ ann

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ [Av1 . . . Avn] = [v1 ⋯ vn]

⎡⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ an1⋮ ⋮

a1n ⋯ ann

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ A (V ) = V A

Ô⇒ [Aw1 . . . Awn] = [w1 ⋯wn]

⎡⎢⎢⎢⎢⎢⎢⎣

a′11 ⋯ a′n1⋮ ⋮

a′1n ⋯ a′nn

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ A (W ) =WA′

16

Effects of changing bases on the representations of A

[AvD1 . . . AvD

n ] = [vC1 ⋯ vC

n ]

⎡⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ an1⋮ ⋮

a1n ⋯ ann

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ [Av1 . . . Avn] = [v1 ⋯ vn]

⎡⎢⎢⎢⎢⎢⎢⎣

a11 ⋯ an1⋮ ⋮

a1n ⋯ ann

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ A (V ) = V A

Ô⇒ [Aw1 . . . Awn] = [w1 ⋯wn]

⎡⎢⎢⎢⎢⎢⎢⎣

a′11 ⋯ a′n1⋮ ⋮

a′1n ⋯ a′nn

⎤⎥⎥⎥⎥⎥⎥⎦

Ô⇒ A (W ) =WA′

16

Effects of changing bases on the representations of A

A (V ) = V A A (W ) =WA′ V =WΓv→w W = V Γw→v

WA′ = A (W ) = A (V Γw→v) = A (V )Γw→v = V AΓw→v =WΓv→wAΓw→v

⇓A′ = Γv→wAΓw→v

Convenient notation:Γv→w = T A′ = TAT −1

17

Effects of changing bases on the representations of A

A (V ) = V A A (W ) =WA′ V =WΓv→w W = V Γw→v

WA′ = A (W ) = A (V Γw→v) = A (V )Γw→v = V AΓw→v =WΓv→wAΓw→v

⇓A′ = Γv→wAΓw→v

Convenient notation:Γv→w = T A′ = TAT −1

17

Effects of changing bases on the representations of A

A (V ) = V A A (W ) =WA′ V =WΓv→w W = V Γw→v

WA′ = A (W ) = A (V Γw→v) = A (V )Γw→v = V AΓw→v =WΓv→wAΓw→v

⇓A′ = Γv→wAΓw→v

Convenient notation:Γv→w = T A′ = TAT −1

17

Example

V = [2 11 2] A = [0.5 1

0 0.5] W = [3 1

1 3]

Workflow:1 find how to express V in terms of W

2 invert, so to find how to express W in terms of V

3 find A′ as “how to express V in terms of W” times A times “how to express W interms of V ”

18

Example

V = [2 11 2] A = [0.5 1

0 0.5] W = [3 1

1 3]

Workflow:1 find how to express V in terms of W

2 invert, so to find how to express W in terms of V

3 find A′ as “how to express V in terms of W” times A times “how to express W interms of V ”

18

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

Extremely important facts!!!

Assume T to be a generic change of basis. Then:

1 eigenvectors and eigenvalues depend only on A:λi eigenvalue of A ⇔ λi eigenvalue of A′ = TAT −1

2 characteristic polynomials depend only on A:

det (λI −A) =p

∏i=1(λ − λi)µ(λi) = det (λI − TAT −1)

3 (corollary) algebraic multiplicities depend only on A

4 eigenspaces depend only on A:ker (λiI −A) = ker (λiI − TAT −1)

5 (corollary) geometric multiplicities depend only on A

19

next lesson: connect algebraic and geometric multiplicities

20