CONTROLLABILITY. Chapter Reachable Set and Controllability. Suppose we have a linear system described by the state equation


 Augustine Curtis
 1 years ago
 Views:
Transcription
1 Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in R n, does there exist a time t, < t < and a piecewise continuous control input u, defined on [,t, such that the solution of () x(t) x? We shall refer to this problem as the reachability problem If for x, there exists an input u which solves the reachability problem, we then say that the state x is reachable from at time t We refer to such an input u as the control which transfers the state from the origin to x at time t Denote the set of states reachable from by R If R R n, then every state is reachable from by suitable control One motivation for studying reachability is the interception or rendezvous problem Suppose you know the trajectory of some vehicle and you want to maneuver your own vehicle to intercept it If R R n, the interception problem is solvable To prepare for the discussion later in the chapter, let us give a slightly more abstract but compact description of the reachability problem Using the variation of parameters formula, x is reachable from at time t if there exists an input u such that x e A(t τ) Bu(τ)dτ (22) Fix the time t Let U be the space of all input signals defined on [ t Define the linear operator L which maps U to R n by Lu e A(t τ) Bu(τ)dτ (23) Recall that if A is an m n matrix (a linear transformation) mapping R n into R m, the range of A, denoted R(A), is the set {y R m y Ax, for some x R n } Analogously, R(L), the range of L, is defined to be the set {x R n x Lu, for some u U} Using this formulation, x is reachable from if and only if x R(L) There is nothing particularly special about the initial state x In fact, if R R n, then every state x is reachable from any initial state x To see this, write the variation of parameters formula for the solution of (2) in the form x(t) e At x 27 e A(t τ) Bu(τ)dτ (24)
2 28 CHAPTER 2 CONTROLLABILITY We see that x reachable from x is equivalent to x e At x is reachable from We now define controllability for the system described by (2) Definition: The system (2) is said to be controllable at time t if for every vector x R n and every initial condition x, there exists an input u which transfers the state from x to x at time t In view of the preceding discussion, studying the controllability property is equivalent to studying whether R R n Example: Consider the following network + R i i 2 R 2 v i  R 3 + C v  v C 2 Figure 2: An RC bridge circuit It is straightforward to verify, using Kirchhoff s current law, that v, v 2 satisfy the following differential equations: [ In state space form, denoting v v v ( C R + C v v v 2 R 3 + v i v R C 2 v 2 v v 2 R 3 + v i v 2 R 2 v 2 T, we have C R 3 ) C R 3 C 2 R 3 ( C 2 R 2 + C 2 R 3 ) v + C R C 2 R 2 Put α C R, α 2 C 2 R 2, β C R 3, β 2 C 2 R 3 We can write the differential equation in the form v [ (α + β ) β v + β 2 (α 2 + β 2 ) Notice that if α α 2, ie, C R C 2 R 2, the difference v v 2 satisfies [ α α 2 d dt (v v 2 ) (α + β + β 2 )(v v 2 ) v i v i (25) which is a homogeneous equation with no input In this case, we cannot manipulate v and v 2 arbitrarily, so that the reachable set is not the whole R 2 To study the reachability problem, we introduce 2 important matrices arising from the system (2)
3 2 REACHABLE SET AND CONTROLLABILITY 29 (i) For t >, define the controllability Gramian at time t W t e Aτ BB T e AT τ dτ (26) e A(t s) BB T e AT (t s) ds (27) Finding W t requires computation of e At and integration Example: A (si A) [ 2 [ s + s + 2 [ s+ (s+)(s+2) s+2 [ s + 2 s + (s + )(s + 2) [ s+ s+ s+2 s+2 [ e At L (si A) e t e e 2t e 2t [ B [ e τ e W t e 2τ [ [ e 2τ [ [ e τ e e 2τ [ e 2τ e τ e 2τ e 2τ [ e 2τ 2e 3τ + e 4τ e 3τ e 4τ e τ e τ e 2τ e 2τ dτ dτ dτ e 3τ e 4τ e 4τ [ 2 ( e 2t ) 2 3 ( e 3t ) + 4 ( e 4t ) 3 ( e 3t ) 4 ( e 4t ) 3 ( e 3t ) 4 ( e 4t ) 4 ( e 4t ) Note that W t is a symmetric n n matrix (ii) Let C AB denote the n nm matrix C AB [B AB A 2 B A n B (28) C AB is called the controllability matrix For m, C AB is a square matrix For m >, C AB is a rectangular, wide matrix Recall that the rank of a p q matrix A is the number of independent columns of A, which is the same as the number of independent rows of A Note that rank(c AB ) n If rank(c AB ) n, we say C AB has full rank Let us first illustrate the computation of C AB in some examples
4 3 CHAPTER 2 CONTROLLABILITY Example : For the circuit described by Figure 2, we have C AB α (α 2 + α β ) + α 2 β α 2 (α α 2β 2 ) + α β 2 Note that det C AB (α α 2 + α β 2 + α 2 β )(α α 2 ) so that C AB is singular (equivalently C AB does not have full rank) if and only if α α 2 Example 2: Consider the following system of 2 carts connected by a spring y u 2 K u M M 2 y 2 Figure 22: A coupled 2cart system Newton s law gives M ÿ K(y y 2 ) + u M 2 ÿ 2 K(y 2 y ) + u 2 [ T Let us recast the equations in state space form Define x y ẏ y 2 ẏ 2 Then ẋ K K M M K M 2 K M 2 x x 2 x 3 x 4 + M M 2 u u 2 The controllability matrix for this system is given by K M M 2 K K M C AB M 2 M M 2 K M 2 M M 2 K M 2 M M 2 K M 2 2 K M M 2 K M 2 2 B AB A 2 B A 3 B Note that C AB has 4 linearly independent columns so that it is full rank Before we discuss the main result of Chapter 2, let us review some geometric ideas from linear algebra Suppose V is a subspace of R n The orthogonal complement of V, denoted by V, is the set of all vectors w such that w T v for all v V
5 2 REACHABLE SET AND CONTROLLABILITY 3 Example: Suppose V is the x y plane, a subspace of R 3 Then V is the zaxis Some important properties of orthogonal complements are: V is a subspace 2 The only vector that lies in both V and V is the zero vector 3 Every vector x R n can be uniquely decomposed as x u + w with u V and w V We refer to the decomposition as R n is the direct sum of V and V, written as R n V V 4 (V ) V 5 Let V and W be subspaces of R n V W if and only if V W, and V W if and only if W V The following is the main result of this chapter It shows how C AB and W t feature in the solution of the reachability problem Theorem 2: R(L) R(W t ) R(C AB ) Proof: The theorem will be proved using the following 2 steps () Prove R(L) R(W t ) (2) Prove R(C AB ) R(W t ) Step : Prove R(L) R(W t ) Using property (5) of orthogonal complements, proving R(L) R(W t ) is equivalent to proving R(L) R(W t ) (a) R(L) R(W t ) : Let v R(L) Then for all u U, This is true for all u if and only if v T e A(t τ) Bu(τ)dτ v T e Aτ B, all τ, τ t But then so that v R(W t ) v T W t v T e Aτ BB T e AT τ dτ (b) R(W t ) R(L) : Let v R(W t ) Then, in particular, v T W t v But v T W t v v T e Aτ BB T e AT τ dτ v v T e Aτ B 2 dτ Thus v T W t v holds if and only if v T e Aτ B, τ t This implies v R(L)
6 32 CHAPTER 2 CONTROLLABILITY This completes Step of the proof Note that in the proof, we have also shown that, for t >, v T e Aτ B, τ t if and only if v T W t Step 2: Prove R(C AB ) R(W t ) First note that we can express e At in the form e At ϕ (t)i + ϕ (t)a + + ϕ n (t)a n (29) for certain functions {ϕ i (t)} An outline of the proof of this result is given in the problem sets Intutively, we can see that this is a consequence of the CayleyHamilton Theorem, which states: then If an n n matrix A has characteristic polynomial p(s) given by p(s) s n + p s n + + p n s + p n p(a) A n + p A n + + p n A + p n I CayleyHamilton Theorem implies A k,k n is expressible as a linear combination of A j, j n This means that the infinite series that defines e At should be expressible in the form of (29) We can now carry out the proof of Step 2 We first show v T C AB if and only if for t > Suppose v T C AB Then v T e Aτ B for all τ, τ t v T e Aτ B v T [ϕ (τ)i + + ϕ n (τ)a n B ϕ (τ)i v T [B AB A n ϕ (τ)i B ϕ n (τ)i Conversely, suppose v T e Aτ B for all τ t Setting τ gives v T B For k,,n, take the kth derivative of v T e Aτ B with respect to τ and evaluate the result at τ This gives successively v T AB v T A 2 B v T A n B so that v T C AB
7 2 REACHABLE SET AND CONTROLLABILITY 33 Now in the proof of Step, we have shown that v T e Aτ B, τ t holds if and only if holds This allows us to conclude that if and only if v T W t v T C AB v T W t Therefore v R(C AB ) if and only if v R(W t ) Hence R(C AB ) R(W t ), which is equivalent to R(C AB ) R(W t ) This completes the proof of Step 2 and the proof of the Theorem Observe that since R(C AB ) is independent ot t, Theorem 2 implies that the reachable set and reachability properties are in fact independent of t We can therefore talk about controllability without reference to t Furthermore, note that for an n n matrix M, R(M) R n if and only if M is nonsingular, and for an n nm matrix N, R(N) R n if and only rank(n) n Combining these observations and the Theorem, we obtain the following Corollary: The following statements are equivalent: (a) R R n (b) W t is nonsingular for any t > (c) C AB has rank n (full rank) Since the linear system (2) is controllable if and only if R R n, the equivalence of statements (a) and (c) of the Corollary can be restated as the following important theorem Theorem 22: The linear system (2) is controllable if and only if Rank [B AB A n B n Theorem 22 allows us to check the controllability property using given data A and B It is a very important result Based on the Theorem 22, we are justified in saying: The pair (A,B) is controllable if Rank[B AB A n B n Suppose x is reachable at time t from x It is not difficult to write down explicitly he control that achieves the transfer from x to x at time t In fact, a control input that achieves the transfer can be verified to be given by u(τ) B T e AT (t τ) ξ (2) where ξ is the solution of W t ξ x e At x If (A,B) is controllable, Wt exists, and (2) can be written explicitly as u(τ) B T e AT (t τ) Wt (x e At x ) (2) Note also that if (A,B) is controllable, the state transfer can be accomplished for any t >, no matter how small, since W t is nonsingular for any t > However, if t is small, Wt will be large, and the control input to achieve the transfer will be large too
8 34 CHAPTER 2 CONTROLLABILITY 22 Some Properties of Controllability In this chapter, we discuss 2 properties of controllability: invariance under a change of basis and invariance under state feedback We also introduce the controllable canonical form for single input systems, which will be very useful for the pole assignment problem discussed in the next chapter () Invariance under change of basis: Recall that if x is a state vector, so is V x for any nonsingular matrix V In fact, if we let z V x, ż V x V AV z + V Bu so that with z as the state vector, the system matrices change from (A,B) to (V AV,V B) We refer to this as a change of basis because if we let the columns of V form a new basis for R n, z is then the representation of x in this new basis (See the Appendix to Chapter 2 for more details on change of basis) We call the transformation V featured in the change of basis a similarity transformation Theorem 23: (A,B) is controllable if and only if (V AV,V B) is controllable for every nonsingular V Proof: C V AV,V B [V B V AV V B V [B AB V C AB Since Rank(V C AB ) Rank(C AB ) the result is proved (2) Invariance under state feedback: A control law of the form u(t) Kx(t) + v(t) (22) with v(t) a new input, is referred to as state feedback The closedloop system equation is given by ẋ (A BK)x(t) + Bv(t) It is an important property that controllability is also unaffected by state feedback Theorem 24: (A,B) is controllable if and only if (A BK,B) is controllable for all K A proof of Theorem 24 is discussed in the problem sets Intuitively, the result seems reasonable since the state feedback law (22) can be reversed by setting v Kx + u Therefore, the ability to control the system using the input v should be the same as that of using input u
9 23 DECOMPOSITION INTO CONTROLLABLE AND UNCONTROLLABLE PARTS Decomposition into Controllable and Uncontrollable Parts If the pair (A, B) is not controllable, there is a particular basis in which the controllable and uncontrollable parts are displayed transparently We illustrate the choice of basis and the computation involved using the circuit example in Section 2 Example: We know from Section 2, that the circuit described by Figure is not controllable if R C R 2 C 2 Using the notation from Section 2, and setting α R C R 2 C 2, we can rewrite (25) as [ [ (α + β ) β v α v + v β 2 (α + β 2 ) α i (23) The controllability matrix is given by [ α α 2 C AB α α 2 so that s [ T is a basis for R(CAB ) Pick a second vector s 2 so that s and s 2 form a basis for R 2, for example, s 2 [ T Set V [ s s2 We then have and V AV [ [ [ [ (α + β ) β β 2 (α + β 2 ) [ [ α β α (α + β 2 ) [ α β (α + β + β 2 ) V B [ [ α α [ α Note that the change of basis produces the following system equation ż Ãz + Bu with [Ã Ã V Ã AV 2 Ã 22 [ B B V B (24) (25) The part corresponding to (Ã, B ), with state component z, is controllable, while the part corresponding to the Ã22, with state component z 2, is uncontrollable, since the z 2 equation is decoupled from z, and z 2 is unaffected by u In general, whenever C AB is not full rank, we can find a basis so that in the new basis, A and B take the form given in (24) and (25), respectively, with (Ã, B ) controllable The procedure is: Find a basis for R(C AB ) Denote the vectors in this basis by {v,v 2,,v q }, where q rank(c AB )
10 36 CHAPTER 2 CONTROLLABILITY 2 Select an additional n q linearly independent vectors v q+ v n so that {v,v 2, v n } form a basis for R n Define the matrix V by V [ v v 2 v n 3 Compute Ã V AV and B V B Ã will take the form (24) and B will take the form (25) For more details, the appendix to Chapter 2 contains a formal proof that the above procedure yields Ã and B as described 24 The PBH Test for Controllability While we normally check controllability by determining the rank of C AB, there is an alternative useful test for controllability, referred to as the PBH test We state the result as a theorem Theorem 25 (PBH Test): (A,B) is controllable if and only if Rank[A λi B n for all eigenvalues λ of A Proof: The statement is equivalent to: (A,B) is not controllable if and only if rank[a λi B < n for some eigenvalue λ of A (i) We first prove rank[a λi B < n implies (A, B) is not controllable Suppose rank[a λi B < n for some eigenvalue λ, possibly complex Thus there exists a vector x, possibly complex such that x [A λi B where x is complex conjugate transpose of x This results in x A λx and Then and x B x AB λx B x A k B λ k x B Thus x [B ABA n B so that [Re x [B ABA n B and (A,B) is not controllable (ii) We now show (A,B) not controllable implies rank[a λi B < n for some eigenvalue λ of A Assume (A,B) is not controllable By the results of Section 23, we can find a nonsingular V so that (V AV,V B) (Ã, B), where Ã is of the form (24) and B is of the form (25), with Ã q q and Ã22 n q n q Now note that for any λ an eigenvalue of Ã22, Rank(Ã22 λi) < n q Hence [Ã λi B Rank [ Ã λi Ã 2 B Ã 22 λi < n
11 24 THE PBH TEST FOR CONTROLLABILITY 37 Next observe that Since V and [ V I [ V V [A λi B I are both nonsingular, V [AV λv B [V AV λi V B [Ã λi B Rank[Ã λi B Rank[A λi B < n for λ an eigenvalue of Ã 22 Finally, note that by the structure of Ã, an eigenvalue of Ã 22 is an eigenvalue of Ã Since a change of basis does not change the eigenvalues, λ is also an eigenvalue of A This concludes the proof that (A, B) not controllable implies there exists an eigenvalue of A, λ, such that rank[a λi B < n It is useful to note that Rank[A λi B n for all eigenvalues λ of A if and only if Rank[A λi B n for all complex numbers λ This is because for λ not an eigenvalue of A, Rank(A λi) n We refer to the eigenvalue which causes Rank[A λi B < n as an uncontrollable eigenvalue We will see in Chapter 3 that such an eigenvalue is in some sense fixed and not movable The PBH test is especially useful for checking when a system is not controllable We give a couple of examples to illustrate its use Example : Consider the following (A,B) pair [ 3 2 A 2 B [ This corresponds to the bridge circuit example withe the following parameter values: α α 2, β 2, and β 2 We know that since α α 2, this system is not controllable We can also verify that the controllability matrix is [ C AB which is singular Let us apply the PBH test First det(si A) s 2 + 5s + 4 so that the eigenvalues are 4 and For the eignenvalue, PBH test gives [ [ 2 2 A ( )I B which has rank 2 On the other hand, for the eigenvalue 4, PHB test gives [ [ 2 A ( 4)I B 2 which has rank By the PBH test, the system is not controllable and 4 is an uncontrollable eigenvalue Example 2: Consider the following (A,B) pair A B 2 2 2
12 38 CHAPTER 2 CONTROLLABILITY By direct expansion along the first row of (si A), we find s det(si A) det s s 2 2 s s 2 (s 2 + 2) + det s 2 2 s s 2 (s 2 + 2) + s 2 s 4 + 3s 2 By inspection, we immediately see that with the eigenvalue, Rank[A (A,B) pair is not controllable B < 4 By the PBH test, this 25 Controllable Canonical Form for SingleInput Controllable Systems For single input systems, there is a special form of system matrices for which controllability always holds This special form is referred to as the controllable canonical form Using a lower case b to indicate explicitly that the input matrix is a column vector for a single input system, the controllable canonical form is given by A (26) α n α n α b (27) It is easy to verify that the controllability matrix for this pair (A,b) always has rank n, regardless of the values of the coefficients α j Hence the name controllable canonical form An A matrix taking the form given in (26) is referred to as a companion form matrix It is straightforward to show that the characteristic polynomial of the companion form matrix is given by det(si A) s n + α s n + + α n The description of the controllable realization in Section 7 reflects in effect the properties of the pair (A,b) given by (26) and (27) It will be seen in the next chapter that for applications to pole assignment in singleinput systems, the controllable canonical form is particularly convenient for control design To prepare for that discussion, we now show that if (A,b) is controllable, but not in controllable canonical form, we can always find a similarity transformation V so that (V AV,V b) will be in controllable canonical form
13 25 CONTROLLABLE CANONICAL FORM FOR SINGLEINPUT CONTROLLABLE SYSTEMS 39 Consider the matrix V [A n b A n 2 b b [b A n b [v v n α α 2 α n 2 α n α n 2 α α n α n 2 α α n 2 α α By controllability, V exists, so that its columns v,,v n form a basis of R n Note that v v 2 v n v n A n b + α A n 2 b + + α n b A n 2 b + α A n 3 b + + α n 2 b Ab + α b b and that Av A n b + + α n Ab + α n b α n b α n b by the CayleyHamilton Theorem α n v n Av 2 v α n v n Av n v n α v n Thus the matrix representation of A with respect to the basis {v v n } looks like [A v α n α n α Similarly, the vector b looks like [b v
14 4 CHAPTER 2 CONTROLLABILITY But [A v and [b v are then related to the original matrices through [A v [b v V AV V b so that they are related by a similarity transformation Thus the new system z(t) V x(t) will satisfy an equation of the form ż A c z + b c u with (A c,b c ) in controllable canonical form
15 APPENDIX TO CHAPTER 2 4 Appendix to Chapter 2: Change of Basis and System Decomposition Let us first quickly review the operation of a change of basis in linear algebra and how it affects the representation of vectors and linear transformations Let E {e,,e n } and V {v,,v n } be two bases in R n Any vector x R n can be represented with respect to either the basis E or the basis V Thus we can write x ξ i e i i η i v i The coefficient ξ i is the ith coordinate of x with respect to the E basis To emphasize this dependence on the basis, we write [x E [ξ ξ 2 ξ n T Now let A be a linear transformation mapping R n to R n Its action on on the basis vector e i can be represented by Ae i a ki e k k The coefficients {a ij } are then the ijth element of the matrix reprentation of A with respect to the basis E We write this as [A E {a ij } Similarly, if Av i i β ki v k k then [A V {β ij } We now describe the operation of a change of basis Suppose we wish to change the basis for R n from E to V Define the linear transformation P by Pe i v i so that {α ij } is the matrix of P in the basis E Then α ki e k k x Hence we have which can be written as or ξ i e i i η i Pe i i ξ k k i η i α ki e k α ki η i i [x E [P E [x V [x V [P E [x E [ α ki η i e k k i
16 42 CHAPTER 2 CONTROLLABILITY For linear transformations, we have Av i β ki v k β ki Pe k β ki α jk e j k k k j but also Av i APe i A α ki e k α ki a jk e j k k j On comparing the two representations for Av i, we get α jk β ki a jk α ki In matrix notation, this corresponds to k k so that [P E [A V [A E [P E [A V [P E [A E[P E We are now ready to describe the details of system decomposition for an uncontrollable pair (A, B), described in Section 23 Let Rank[B ABA n B q < n, and let {v v q } be a basis for Range[B ABA n B M We can pick additional basis vectors {v q+,,v n } so that {v,,v q,v q+,,v n } form a basis for R n Now observe that by the CayleyHamilton Theorem, AM M Hence for i q, q Av i α ji v j for some scalars α ji, j,,q With respect to the basis V {v,,v n }, A takes the form [ [A V Ã Ã Ã 2 Ã 22 Similarly the ith column of B, Thus, we obtain and b i j q β ji v j j V AV Ã [ Ã Ã 2 Ã 22 V B B where Ã is a q q matrix and B is a q m matrix We claim that (Ã, B ) is controllable For, [ Rank[ B Ã B B Ã Ã n B Rank B Ã n B q But by the CayleyHamilton theorem, for k, Ã q+k B is linearly dependent on Ãj B, j,,q Hence Rank[ B Ã B Ã q B q so that (Ã, B ) is controllable [ B
Recall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationLinear Control Systems
Chapter 3 Linear Control Systems Topics : 1. Controllability 2. Observability 3. Linear Feedback 4. Realization Theory Copyright c Claudiu C. Remsing, 26. All rights reserved. 7 C.C. Remsing 71 Intuitively,
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationUsing determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:
Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationMATH36001 Background Material 2015
MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationMatrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More information= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are
This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In
More informationWHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?
WHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course
More informationChapter 1  Matrices & Determinants
Chapter 1  Matrices & Determinants Arthur Cayley (August 16, 1821  January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed
More informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationDeterminants. Dr. Doreen De Leon Math 152, Fall 2015
Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationEE 580 Linear Control Systems VI. State Transition Matrix
EE 580 Linear Control Systems VI. State Transition Matrix Department of Electrical Engineering Pennsylvania State University Fall 2010 6.1 Introduction Typical signal spaces are (infinitedimensional vector
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More informationUsing the Singular Value Decomposition
Using the Singular Value Decomposition Emmett J. Ientilucci Chester F. Carlson Center for Imaging Science Rochester Institute of Technology emmett@cis.rit.edu May 9, 003 Abstract This report introduces
More informationDeterminants. Chapter Properties of the Determinant
Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation
More information22 Matrix exponent. Equal eigenvalues
22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to
More informationVarious Symmetries in Matrix Theory with Application to Modeling Dynamic Systems
Various Symmetries in Matrix Theory with Application to Modeling Dynamic Systems A Aghili Ashtiani a, P Raja b, S K Y Nikravesh a a Department of Electrical Engineering, Amirkabir University of Technology,
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationSergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014
Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of
More information3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationChapter 12 Modal Decomposition of StateSpace Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationLinear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.
Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a subvector space of V[n,q]. If the subspace of V[n,q]
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationB such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix
Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.
More informationINTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More information18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 206 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are righthandsides b for which A x = b has no solution (a) What
More informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More informationMath 315: Linear Algebra Solutions to Midterm Exam I
Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More information2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors
2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the
More informationThe Hadamard Product
The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationSystem of First Order Differential Equations
CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationLecture Notes 1: Matrix Algebra Part B: Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,
More informationLecture 4: Partitioned Matrices and Determinants
Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying
More informationPresentation 3: Eigenvalues and Eigenvectors of a Matrix
Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues
More informationMathematics Notes for Class 12 chapter 3. Matrices
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationHelpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
More informationMatrices, transposes, and inverses
Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrixvector multiplication: two views st perspective: A x is linear combination of columns of A 2 4
More informationPractice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.
Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular
More informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
More information