EE611 Deterministic Systems
Transcript of EE611 Deterministic Systems
EE611Deterministic Systems
Vector Spaces and Basis ChangesKevin D. Donohue
Electrical and Computer EngineeringUniversity of Kentucky
Matrix Vector MultiplicationLet x be an nx1 (column) vector and y be a 1xn (row) vector:Dot (inner) Product: yx=c = |x||y|cos(θ) where c is a scalar (1x1) and θ is angle between y and xProjection: Projection of y onto x is denoted by (yx)x = |y|cos(θ) = yx/|x|Outer Product: xy=A where A is an nxn matrix.Matrix-Vector Multiplication: Let x be an nx1 vector and A be an nxn matrix:
where ' denotes transpose, and vectors denote a row vector partition in the first expression and a column vector partition in the second expression.
A x=[a١
a٢
⋮aN]x=[a١ x
a٢ x⋮
aN x ] x ' A=x ' [a١ a٢ ... aN ]=[x 'a١ x ' a٢ ... x 'aN]
ai
Linear IndependenceConsider an n-dimensional real linear space ℜn. A set of vectors {x1, x2, ... xm}∈ℜn are linearly dependent (l.d.) iff ∃ a set of real numbers {α1,α2, ... αm} not identically equal to 0 ∋
Otherwise the vectors are linearly independent (l.i.)
Show that if a set of vectors are l.d., then at least one of the vectors can be expressed as a linear combination of the others.
The dimension of the linear space is the maximal number of l.i. vectors in the space.
١ x١٢ x٢...m xm=٠
Basis and RepresentationA set of l.i. vectors in ℜn is a basis iff every vector in ℜn can be expressed as a unique linear combination of these vectors.
Given a basis for ℜn {q1, q2, ..., qn}, then every vector in ℜn
can be expressed as:
where is called the representation of
x=١q١٢q٢...n qn=[q١ q٢ ... qn ][١
٢
⋮n]=Q
x
ExampleFind the representation of noted point with orthonormal basis Q in terms of basis P.
q[١٠]=١ q[٠١]=٢ p[١٠]=١ p٢=[ ١٢
١٢ ]
[−٠.٥١.٥ ] [??]
NormsThe generalization of magnitude or length is given by a metric referred to as a norm. Any real valued function qualifies as a norm provided it satisfies:
∥x∥≥٠∀ x and ∥x∥=٠ iff x=٠
∥ x∥=∣∣∥x∥ for any real scalar
Non-negative
Scalable Consistency
∥x١x٢∥≤∥x١∥∥x٢∥ ∀ x١ ,x٢ Triangular Inequality
Popular NormsGiven
The 1-norm is defined by
The 2-norm (Euclidean norm)
The infinite-norm
Why do you think this is called the infinity norm? Hint: What would a 3-norm, … 100-norm look like?
∥x∥١ :=∑i=١
n
∣xi∣
x=[ x١ x٢ ... xn ] '
∥x∥٢:=٢∑i=١
n
x i ٢=x ' x
∥x∥∞ :=maxi∣xi∣
OrthonormalVector x is normalized, iff its Euclidean norm is 1(self dot product is 1). Vectors xi and xj are orthogonal iff their dot product is 0.
A set of (column) vectors {x1, x2, ... xm} are orthonormal iff
xi ' x j={٠ if i≠ j١ if i= j
∀ i , j
OrthonormalizationGiven a set of l.i. vectors {e1, e2, ... en}, the Schmidt orthonormalization procedure can be used to derive an orthonormal set of vectors {q1, q2, ... qn} forming a basis for the same linear space:
u١:=e١ q١:= ١∥u١∥u١
u٢:=e٢−q١' e٢q١ q٢:= ١∥u٢∥u٢
u٣:=e٣−q١' e٣q١−q٢ ' e٣q٢ q٣:= ١∥u٣∥u٣
un :=en−∑k=١
n−١
qk ' enqk qn := ١∥un∥un
Project and subtract Normalize
................................................
Linear Algebraic EquationsConsider a set of m linear equations with n unknowns:
Range space of A is the set all vectors resulting from all possible linear combinations of the columns of A.
The rank of coefficient matrix A ( rank(A) ) is equal to its number of l.i columns (or rows). rank(A) is also denoted as ρ(A)➢If rank(A) = n, a unique solution x exists given any y➢If rank(A) ≤ m < n, many solutions x exist given any y (underdetermined)➢If rank(A) ≤ n < m no solutions x may exist for some y (overdetermined)
y=A x
y١=a١١ x١a١٢ x٢...a١n xn
y٢=a٢١ x١a٢٢ x٢...a٢n xn
...............................ym=am١ x١am٢ x٢...amn xn
y=a١ x١a٢ x٢...an xn ai=[a١i a٢i ... a٣i ] '
NullityThe vector x is a null vector of A iff Ax=0
The null space of A is the set of all null vectors.
The nullity of A is the maximum number of l.i. vectors in its null space (i.e. dimension of null space).
nullity (A) = n - ρ(A)
Conditions for Solution ExistenceGiven mxn matrix A and mx1 vector y, an nx1 solution vector x exists for y=Ax iff y is in range space of A.
Given matrix A, a solution vector x exists for y=Ax, ∀ y iff A is full row rank (ρ(A) = m).
A= [A⋮y ] ⇔ ∃x such that A x=y
∃x such that A x=y ∀ y⇔A =m
Conditions for Unique SolutionGiven mxn matrix A and mx1 vector y, let xp be a solution for y=Ax. If ρ(A) = n (nullity k= 0), then xp is unique, and if nullity k > 0 then for any set of real αi's, x given below is a solution.
where vector set {n1, n2, ... nk} is a basis for the null space.
The above solution is also referred to as a parameterization of all solutions.
x=xp١ n١٢ n٢...k nk
Singular MatrixA square matrix is singular if its determinant is 0.
Given nxn non-singular matrix A, then for every y, a unique solution for y=Ax exists and is given by A-1y=x.
The homogeneous equation 0=Ax has a non zero solution iff A is singular, otherwise x=0 is the only solution.
Change of BasisDenote a representation of x with respect to (wrt) basis as , and representation wrt As . Note that basis vectors are assumed wrt the orthonormal basis. Find a change of basis transformation such that
Show that
{ e١, e٢, ... en}{e١, e٢, ... en}
=P =Q
P=Q−١=E−١E where E=[e١, e٢, ... en ] and E=[ e١, e٢, ... en ]
ei
Similarity TransformationConsider nxn matrix A as a linear operator that maps ℜn into itself. The vector representations are wrt . Determine the new representation of the linear operator wrt basis
Show that: where
The operation that changes the basis of the linear operator using a pre and post multiplication of a matrix and its inverse is referred as a similarity transform.
{ e١, e٢, ... en}{e١, e٢, ... en}
A=P A Q
P=Q−١=E−١E with E=[e١, e٢, ... en ] and E=[ e١, e٢, ... en ]