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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
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