This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 6.241 Dynamic Systems and Control
Lecture 4: Singular Values
Readings: DDV, Chapter 4 Emilio Frazzoli
Aeronautics and Astronautics Massachusetts Institute of Technology February 14, 2011 E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 1/9 Outline 1 Singular Values 2 Norm computations through singular values E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 2/9 Unitary Matrices A square matrix U ∈ Cn×n is unitary if U � U = UU � = I .
A square matrix U ∈ Rn×n is orthogonal if U T U = UU T = I .
Properties:
If U is a unitary matrix, then �Ux �2 = �x �2 , for all x ∈ Cn .
If S = S � is a Hermitian matrix, then there exists a unitary matrix U such that
U � SU is a diagonal matrix. 1
For any matrix A ∈ Rm×n , both A� A ∈ Rn×n , AA� ∈ Rm×m are Hermitian ⇒
can be diagonalized by unitary matrices.
For any matrix A, the eigenvalues of A� A and AA� are always real 2 and
nonnegative 3 (in other words, A� A and AA� are positive deﬁnite). = S � ⇔ �Sx , y � = �x , Sy �. Let v1 be an eigenvector of S , and let M1 = R(v1 )⊥ . If
u ∈ M1 , then so is Su : �Su , v1 � = �u , Sv1 � = �u , λ1 v1 � = 0. All other eigenvectors must b e in
M1 . Finite induction gets the result.
2 Assuming �v , v � = 1, λ = �Sv , v � = �v , Sv � = �Sv , v �� = λ�
11
1
11
1
1
11
1
3 0 < �Av , Av � = v � A� Av = λ v � v .
1
1
1
111
1
1S E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 3/9 Singular Value Decomposition Theorem (SVD)
Any matrix A ∈ Cm×n can be decomposed as A = U ΣV , where U ∈ Cm×m and
V ∈ Cn×n are unitary matrices. The matrix Σ ∈ Rm×n is “diagonal,” with
nonnegative elements on the main diagonal. The nonzero elements of Σ are
√
called the singular values of A, and satisfy σi = ith eigenvalue of A� A.
Proof (assuming rank(A) = m):
Since AA� is Hermitian, there exist a diagonal matrix Λ = diag(λ1 , λ2 , . . . , λm ) > 0 such that U ΛU � = AA� . 2
2
2
Write Λ = Σ2 = diag(σ1 , σ2 , . . . , σm ) 1
�
�
Deﬁne V1 := Σ−1 U � A ∈ Rm×n . Clearly, V1 V1 = Σ−1 U � AA� U Σ−1 = I m×m .
1
1
1 Construct V = [V1 , V2 ] ∈ Cn×n by choosing the columns in V2 so that V is
unitary, and Σ = [Σ1 , 0] ∈ Rn×n , by padding with zeroes.
�
�
Hence, ΣV � = Σ1 V1 + 0V2 = U � A, i.e., A = U ΣV � . E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 4/9 Singular Vectors
If U and V are written as sequences of column vectors, i.e.,
�
�
�
�
U = u1 , u2 , . . . , um and V = v1 , v2 , . . . , vn , then
A = U ΣV � = r
� σi ui vi� i =1 The columns of U are called the left singular vectors, and the columns of V are called the right singular vectors. Note: Ax can be written as the weighted sum of the left singular vectors, where the
weights are given by the projections of x onto the right singular vectors:
Ax = r
� σi ui (vi� x ), i =1 The range of A is given by the span of the ﬁrst r vectors in U The rank of A is given by r ; The nullspace of A is given the span of the last (n − r ) vectors in V . E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 5/9 Outline 1 Singular Values 2 Norm computations through singular values E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 6/9 Induced 2norm computation Theorem (Induced 2norm)
�A�2 = sup
x �=0 �Ax �2
= σmax (A).
�x �2 Proof:
sup
x =0
� �Ax �2
�U ΣV � x �2
�ΣV � x �2
= sup
= sup
=
�x �2
�x �2
�x �2
x �=0
x �=0
��n
�
2
2 1/2
�Σy �2
�Σy �2
i = σ  yi 
sup
= sup
= sup �� 1 i
�1/2 ≤ σmax (A).
n
y =0 �y �2
y =0
y �=0 �Vy �2
�
�
yi 2
i =1 Assuming σmax = σ1 , the supremum is attained for y = (1, 0, . . . , 0). This
corresponds to x = v1 , and Av1 = σ u1 E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 7/9 Minimal ampliﬁcation Theorem
Given A ∈ Cm×n , with rank(A) = n,
�Ax �2
= σn (A).
x �=0 �x �2
inf Proof:
�Ax �2
�U ΣV � x �2
�ΣV � x �2
= inf
= inf
=
x �=0 �x �2
x �=0
x �=0
�x �2
�x �2
��n
�
2
2 1/2
�Σy �2
�Σy �2
i =1 σi yi 
inf
= inf
= inf ��
�1/2 ≥ σmin (A).
n
y =0 �y �2
y �=0
y �=0 �Vy �2
�
yi 2
inf i =1 Assuming σmin = σn , the supremum is attained for y = (0, . . . , 0, 1). This
corresponds to x = vn , and Avn = σ un E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 8/9 Frobenius norm computation Theorem
�A�F = �r
� �1/2
2 σi (A) i =1 Proof:
⎞1/2
⎛
n
m
��
1/2
1/2
aij 2 ⎠ = (Trace(A� A)) = (Trace(V Σ� U � U ΣV � )) =
�A�F = ⎝
j =1 i =1 � �1/2 �
�1/2
Trace(V � V Σ2 )
= Trace(Σ2 )
= �r
� �1/2
σi2 i =1 E. Frazzoli (MIT) Lecture 4: Singular Values Feb 14, 2011 9/9 MIT OpenCourseWare
http://ocw.mit.edu 6.241J / 16.338J Dynamic Systems and Control
Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . ...
View
Full
Document
 Spring '11
 Prof.EmilioFrazzoli

Click to edit the document details