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# Eigenvalues and eigenvectors of a matrix A

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### State space transformations

• Let us consider the following linear time-invariant system:

 ˙x(t) = A x(t) + B u(t)

y(t) = C x(t) + D u(t) (1)

• A state space transformation can be obtained using a biunivocal linear transformation which links the old state vector x with the new vector x:

x = T x where T is a square nonsingular matrix.

• Applying this transformation to system (1) one obtains “a different but equivalent” mathematical description of the given dynamic system:

 ˙x(t) = A x(t) + B u(t) y(t) = C x(t) + Du(t)

• The matrices of the two systems are linked together as follows:

A = T1AT, B = T1B, C = CT

• Properly choosing matrix T it is possible to obtain mathematical descrip- tions of the given system (the canonical forms) characterized by matrices A, B, C and D which have particularly simple structures.

• For each choice of matrice T one obtains a “different but equivalent”

mathematical description of the given system. All these different mathe- matical models maintain the basic physical properties of the given dynamic system: stability, controllability and observability.

• The transformed matrices A, B and C maintain the same geometric and mathematical internal properties of matrices A, B and C of the given system: matrices A and A have the same eigenvalues, the reachability and observability subspaces do not change, etc.).

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### Eigenvalues and eigenvectors of a matrix A

• Let A be a square matrix of dimension n. If there exists a nonzero vector v and a scalar λ such that:

Av = λv ↔ (A − λI)v = 0

then

λ is an eigenvalue of matrix A;

v is an eigenvector of matrix A associated to eigenvalue λ.

A(λ) = det(λI − A) is the characteristic polynomial of matrix A.

A(λ) = 0 is the characteristic equation of matrix A.

• The eigenvalues λi of matrix A are the solutions of the characteristic equation ∆A(λ) = 0.

• The molteplicity ri of the eigenvalue λi as a solution of the characteristic equation is known as algebraic molteplicity of the eigenvalue λi.

• The set of all the eigenvalues λi of matrix A is the spectrum of matrix A.

• Let λi be an eigenvalue of matrix A. The set of all the solutions vi of the system:

(A − λiI) vi = 0

is a vector space called the autospace Uλi associated to the eigenvalue λi.

• The dimension mi of the autospace Uλi is called geometric molteplicity of eigenvalue λi.

• Property. The geometric molteplicity mi is always smaller or equal to the algebraic molteplicity ri: mi ≤ ri.

• Property. “Distinct” eigenvalues λ1, . . . , λh are always associated to “li- near independent” eigenvectors v1, . . . , vh. Two autospaces Uλi and Uλj

associated to distinct eigenvalues λi and λj, are disjoined.

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same eigenvalue λ = 1. This eigenvalue has algebraic molteplicity r = 2 for both the matrices, but it has different geometric molteplocity m:

A =  1 0 0 1



, (λ − 1)2 = 0, r = 2, m = 2

A =  1 1 0 1



, (λ − 1)2 = 0, r = 2, m = 1

• Property. Two similar matrices have the same characteristic polynomial.

Let T be a non singular matrix and let A be the matrix obtained from A applying the similitude transformation:

A = T1AT The characteristic polynomial of matrix A is:

A(λ) = det(λI − A) = det(λI − T1AT)

= det(λT1T − T1AT) = det(T1(λI − A)T)

= det T1 det(λ − A) det T = det(λ − A) = ∆A(λ)

• If the matrices A and A have the same characteristic polynomial, then they also have the same eigenvalues.

• Matlab example:

---Matlab commands --- A=[1 4 2;...

9 2 4;...

1 0 5]; % Matrix A

[V, D]=eig(A) % D = Eigenvalues; V = Eigenvectors;

poles=roots(poly(A))’ % Roots of the characteristic polynomial ---Matlab output---

V =

0.6024 -0.5121 -0.4378 -0.7957 -0.8440 -0.6627 -0.0634 -0.1593 0.6075 D =

-4.4944 0 0

0 8.2151 0

0 0 4.2793

poles =

-4.4944 8.2151 4.2793

---

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Example. Let us consider a mechanical dynamic system composed by a mass M , a spring with stiffness K, a viscous friction b and with an external force F acting on mass M:

F

1

¨ x

- 

1 M

1 s

?

?

?

˙x

 -

 

b

6

6

- -

x

2

 -

6 1 s 6

K

6

-  0

x(t) b

M F K

1 2

The dynamic behavior of the system is described by the following differential equation:

F = M ¨x+ b ˙x + Kx

The transfer function G(s) of the considered dynamic system is:

G(s) = x(s)

F(s) = 1

M s2 + bs + K =

1 M

s2 + Mb s+ MK The poles s1 and s2 of the transfer function G(s) are:

s1 = − b 2M

s

 b 2M

2

K

M, s2 = − b 2M +

s

 b 2M

2

K M

Let x = [x, ˙x]T and y = x. The considered dynamic system can also be described in the state space as follows:

 ˙x

¨ x



=

 0 1

K

M b

M

  x

˙x



+  0

1 M

 F

y = 

1 0 

x The system matrices have the following form:

A =

 0 1

K

M b

M



, b =  0

1 M



, c =  1 0  The characteristic polynomial of matrix A is:

det(λI − A) = λ2 + λ b

M + K M

The zeros λ1 and λ2 of the characteristic polynomial are equal to the eigenvalues of matrix A:

λ1 = − f 2M

s

 f 2M

2

K

M, λ2 = − f

2M + s

 f 2M

2

K M

The eigenvalues λ1 and λ2 of matrix A are equal to the poles s1 and s2 of the transfer function G(s) = c(sI − A)−1b of the considered dynamic system.

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˙x = A x,

let v be an eigenvector of matrix A associated to eigenvalue λ:

(λI − A)v = 0

and let x be a state vector proportional to the eigenvector v:

x = α v ∈ Im[v].

• Under these hypotheses, it follows that the velocity vector ˙x is parallel to x:

˙x = A x = λ x.

• Graphical representation:

x

˙x=λx

x1 x2

Im[v]

a) λ > 0

˙x=λx x

x1 x2

Im[v]

b) λ < 0

• The straight lines associated to real eigenvector v are, within the state space, the only straight line trajectories which can be found in the given autonomous dynamic system.

• If the initial state x0 of a linear dynamic system belongs to the straight line associated to eigenvector v, then the corresponding free evolution of the system belongs to the same straight line.

• If λ > 0 the trajectory moves away from the origin, if λ < 0 trajectory moves toward the origin, and finally if λ = 0 the trajectory coincides with the initial condition: x(t) = x0.

x0

x(t)

x1 x2 a) λ > 0

Im[v]

x0 x(t)

x1 x2 b) λ < 0

Im[v]

Riferimenti

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