Canonical form pdf De nition 2. Structure Theory: The Jordan Canonical Form 18 2. The matrix [T] Bis called a Jordan canonical form of T. Sundar B N. 52 13. But if 1 k r, d k of the form aeλx for some aP R. = 0 Solution: From the transfer function (16), we obtain the following parameters: b 0 = 1, b 1 = 3, b 2 = 3, a 1 = 2, and a 2 = 1. Example: d’Alembert’s solution; Parabolic case. Each of these canonical form has specific advantages which makes it convenient for use in particular design technique. Instructor: TonyPantev Universityof Pennsylvania Math are known as canonical forms. Counting from left to right, the j-th column consists of the p j dots that correspond to the vectors of C Section 3. We will always take transformations Q i= Q i(q;p;t) and P i= P i(q;p;t) to be invertible in any of the canonical variables. Observability Matrix: O(A,C) = C Definition 6 If Lis a nilpotent matrix, a Jordan form of Lis a Jordan matrix J= P−1LP. ) Canonical Forms Minterms and Maxterms A minterm (aka standard product) is an AND term containing all variables. (b) Only one matrix in rational canonical form appears as the matrix of T with respect to some ordered basis of V. They have algebraic multiplicities p 1; ;p r. 11) the derivatives change according to: First Order ux = urrx +ussx, uy = urry +ussy, (3. Similarly, a linear program in standard form can be replaced by a linear program in canonical form by replacing Ax= bby A0x b0where A0= A A and b0 The Jordan canonical form Francisco{Javier Sayas University of Delaware November 22, 2013 The contents of these notes have been translated and slightly modi ed from a previous version in Spanish. Then take each term with a missing variable and Inner Products and Quadratic Forms In fact, any matrix P = [i l] ,with a + b = 2, can replace A without changing the quadratic form Q. [T] Bis a diagonal matrix if and only if each A i is of the A matrix representation of the form [T]fl (from the first slide) is called a rational canonical form of T, and the basis fl is called a rational canonical basis. (1) can be converted into one of three canonical or standard forms, which we call hyperbolic, parabolic or elliptic. 4. 5 The Rational Canonical Form Here pis a monic irreducible factor of the minimum polynomial m T and is not necessarily of degree one. There’s no way it could be genuinely unique, since for any square matrices Cand D(perhaps of di erent sizes), the two matrices C Dand D Care conjugate. 2 Modules Most proofs of the existence of Rational Canonical Form rely on the module iis a square matrix of the form ( ) or a Jordan block J de ned above, such that is an eigenvalue of T. proceed as in Example 1 to obtain u = 0 which is the 7. For multi-input systems, their This is a first course in feedback control of dynamic systems. So, what’s the point of the Jordan canonical form? Quadratic forms 2 With respect to dual bases the matrix M∇ is the matrix of that linear transformation. Canonical Forms 3 vector in B i, and the dots are con gured according to the following rules. Let F p denote the eld constructed earlier in the course, consisting of all matrices of the form f(B);f2F[x], where B= C(p), the companion matrix of p. 5 Canonical Forms In general, the unique algebraic expression for any Boolean function can be obtained from its truth table by using an OR operator to combined 2. characterisitic value is also known an eigen value. iii. 48 12. Dr. Theorem 5 tells us that Jordan form is unique up to ordering of the blocks Ji. The document discusses different canonical state-space forms for representing linear systems, including the controllable canonical form, consider transformations to canonical form. In this section of the course, we explore canonical forms with three main types of results: 1. Tall-Respondek [35] solved the problem of canonical form for single-input and linearly controllable systems. hiremath@iitj. Chapter 7. The course description reads as follows: Jordan Canonical Forms December 6, 2006 1 Introduction We know that not every n n matrix A can be diagonalized. For example x = (x 1,x 2,x 3) and (A,b) = 1 1 1 5 1 2 0 4! → B = 0 −1 1 1 1 2 0 4! then it is a canonical form A canonical form of a linear transformation is a matrix representation in a basis chosen to make that representation simple in form. Here we will discuss basic facts. 2 Compute n (k) i = rk(A iI)k for 1 i rand 0 k p i(you can actually stop before p Canonical Forms Canonical forms are the standard forms of state space models. Canonical Forms: Now the problem is to determine u and v so that the equation (7) takes the simplest form. A major theme in the analysis Notes on the Jordan canonical form Introduction Let Vbe a finite-dimensional vector space over a field F, and let T : V! Vbe a linear operator such that (T-a 1)k 1 (T-a m)km= 0, (1) for a 1,,a mdistinct numbers in F. Presentations and the canonical form for modules. Consider the system in Modal Canonical Form x = " 1 1 0 0 0 1 0 0 0 0 2 0 0 0 0 3 # x + 1 0 0 2 u y = [1 0 1 1]x From the example seen in the Tutorial, Controllability and Observability in Modal Form equations, we see that the r st 1 is controllable and observable 2 is not controllable, although observable 3 is controllable and observable Examples and standard form Fundamental theorem Simplex algorithm Canonical form I First suppose the standard form is Ax = b, x ≥ 0 I One canonical form is to transfer a coefficient submatrix into I m with Gaussian elimination. Therefore, it is impossible to put this matrix in Jordan Canonical Form. Thus we see that the Bernoulli distribution is an exponential family distribution with: η = π 1−π (8. t/: Then CDŒB AB An1B D 2 6 4 1 0 0 0 ::: 0 0 0 1 3 7 5DIn: This is why it is called controllability form! If a system is controllable, we can instantaneously move the state from any known state to any other Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. 3. Thus, Jordan Canonical Form can only be achieved for matrices in an algebraically closed eld, which leads us to a second canonical form: that is, Rational Canonical Form. Reduce the equation uxx +2uxy +uyy = 0 to canonical form. 2. Computer-based analysis, combined with a modern accompanying laboratory, provide a realistic setting for mastering several important design methodologies. form is unique under transformation of arbitrary degree, it is call a canonical. As there are three types of canonical forms, hyperbolic, parabolic and elliptic, we will deal with each type separately. So, we get, * 2 2 * 2 2 0, 0, x x y y x x y y a au bu u cu c av bv v cv Or, equivalently, Canonical Form - Free download as PDF File (. 2 Canonical Form. Find a T-invariant subspace U V satisfying V = W U. However, if we insist that the matrix P shouId be symmetric, then we must have a =b; and hence, the choice is unique, namely, [: :I. The generalized eigenvectors of eigenvalue λ are the functions of the form ppxqeλx for some polynomial ppxq. Bhuvaneswari BMS College for Women October 2020 RB (BMSCW) M304T October 20201/35. 5. A linear program with n variables is in canonical form if it is of the following form max ~c>~x A~x =~b ~x 0 where A = (a 4. We also have another simple case for which b2 −4ac >0 condition is satisfied. The corresponding Jordan canonical form is J= A 1 0 0 A 2 ; where A 1 = 2 0 0 2 and A 2 = 4 1 0 4 . t/C 2 6 4 1 0 0 3 7 5u. Shu It is our task in this chapter to find a simple form or a specific form (i. , canonical form) of a matrix as a result of the similarity transformation. ii) The subset Z[i] = fa + ib 2C : a;b 2Zgis easily checked to be a ring under Lecture notes Partial differential equations. Hence, (A I)2v = 0 for any generalized 1-eigenvector v. There are several canonical forms of state space models –Phase variable canonical form –Controllable Canonical form –Observable Canonical form STANDARD FORMS MINTERM AND MAXTERM EXP. Example 2. The first decomposition leads to therational canonical form. Peet Lecture 7: Controllability 17 / 17. Kirankumar R. To do this, we transform the Jordan canonical form what if A cannot be diagonalized? any matrix A ∈ Rn×n can be put in Jordan canonical form by a similarity transformation, i. Indeed, given any prescribed An matrix J is said to be in Jordan canonical form if it is a matrix of the form where each is either a diagonal matrix or a Jordan block matrix. As a consequence, for every 1 6 i 6 r, the vectors {xkϕ(ηi) : 0 6 k 6 ni −1} (10) form a basis of F[x]ϕ(ηi), relative to which the matrix of the restriction of LA is the companion matrix of di. The Jordan canonical form (Lecture 21) 26 Chapter 3. However, it turns out that we can always put matrices A into something called Jordan Canonical Form, which means that A can be written as A = B 1 2 6 6 6 4 J1 J2 Jk 3 7 7 7 5 B; where the Ji are certain block matrices of the form 6. 1) can be reduced to an equation of one of Canonical Forms • Truth table is the unique signature of a 0/1 function • The same truth table can have many gate realizations – We’ve seen this already – Depends on how good we are at Boolean simplification • Canonical forms – Standard forms for a Boolean expression – We all produce the same expression 4. T−1AT = J = J1 Jq where Ji = λi 1 Notes on the Jordan canonical form Introduction Let Vbe a finite-dimensional vector space over a field F, and let T : V! Vbe a linear operator such that (T-a 1)k 1 (T-a m)km= 0, (1) for a 1,,a CANONICAL FORMS IN LINEAR ALGEBRA Let kbe a eld, let V be a nite-dimensional vector space over k, and let T: V ! V be an endomorphism. proof Suppose Ais equivalent to a matrix Bin Smith canonical form. THEOREM 1. t/ D h ˇ1 ˇ2 ˇ3 i x. However, we might lack canonical coordinates. In this case, A= 1, B= 2, C= 1. For this purpose, we should first find eigenvalue(s) and corresponding eigenvector(s) of the matrix. Linear algebra teaches us, laboriously, The canonical form is u +u = − 1 2α u . Upon identifying withVbb, theform ∇ is symmetric if and only if the map is equal to its own Canonical forms for matrices over a Euclidean Domain. While the classification is similar to that of conic sections in plane geometry, the canon-ical forms are similar to their standard forms. txt) or read online for free. 56 poonen/papers/ring. ) Let V be a vector space over a ̄eld F and T 2 L(V; V ) be linear operator. A minterm, denoted as mi, where 0 ≤ i < 2n, is a product (AND) of upper-triangular (in fact, it is in Jordan canonical form). Nilpotent maps and Jordan blocks (Lectures 19-20) 24 2. The purpose of this note is to explain how to find a Jordan basis, i. PDEs are classified as hyperbolic, parabolic, or elliptic based on the 4. Application to rational and Jordan canonical forms. 3. t/ y. That is, a Jordan matrix is a matrix with Jordan blocks down the diagonal and zeros everywhere else. ac. Written in new variables ξ and η, the three forms are: uξξ - uηη + . pdf), Text File (. The three equations in Example 1 above are of particular interest not only because they are derived from physical principles, but also because every second-order linear equation of the form (4. Three Canonical or Standard Forms of PDE's Every linear 2nd-order PDE in 2 independent variables, i. That is, B= 2 6 6 6 4 f 1 f r 0 0 0 3 7 7 7 5 and f 1 jf 2 jj f r: Then r= ˆ(A), the determinantal rank of A. For spaces without and with inner product structure, the equivalence relations and canonical forms associated with an operation on Vof dimension nand Wof dimension nare shown in Figure1. The array consists of n i columns (one column for each cycle). We will choose the new variables u and v such that the coefficients a* and c* in equation (6) vanish. 7) Along these directions the partial differential equation takes a simple form called Nor-mal or Canonical form. We first suppose that a ≠ 0, b ≠ 0, c ≠ 0. 1 Uniqueness of the Smith Canonical Form THEOREM 6. 5 Canonical Forms In general, the unique algebraic expression for any Boolean function can be obtained from its truth table by using an OR operator to combined all minterms for which the function is equal to 1. Generalized eigenspaces and Cayley–Hamilton (Lectures 17-18) 21 2. The resulting state-space model in controllable canonical form is obtained as IntroductionClassificationsCanonical formsSeparation of variables Introduction to Partial Differential Equations Introductory Course on Multiphysics Modelling Computing the Jordan Canonical Form Let A be an n by n square matrix. A design-oriented approach is stressed. For example the KdV equation can be described with a Hamiltonian and Poisson bracket but there isn’t a pair of canonical coordinates. After beginning with background material, we introduce Jordan Canonical Form and related Download book PDF. Solution. 1 (De ̄nition. . We say that the ordered basis Bis a Jordan canonical basis for T. The solution of dy dx −1 = 0 is x−y= c1 Take ξ= x−y. (We saw that if degp= n, then F p= fa 0I n+ + a n 1B n 1ja 0 Canonical form and Standard form of LPP - Download as a PDF or view online for free. . r. The minimal polynomial (Lecture 16) 19 2. Putting these bases together brings T to the desired form. De nition 1. 6) dy dx = B − √ B2 − 4AC 2A (7. 7) T(x) = x (8. 3 Normal or Canonical Form For hyperbolic equations, there exists two real directions, called characteristic direc-tions given by dy dx = B + √ B2 − 4AC 2A (7. tion 3. • Given the factorizations (3), for every 1 6 i 6 r one . 1 To compute a Jordan canonical form of a matrix Ait su ces to follow these steps: 1 Compute the distinct eigenvalues of A: 1; ; r. The equation α2 +2α+1 = 0 has equal roots λ= −1. a basis for Vsuch that [T] EXAMPLE: Controllability canonical form: x. (C) uxy +ux +uy = 2x. Practice Problems 1. Hiremath (k. VerifythatCo(sI Ao) 1Bo = G(s). 8) A(η) = −log(1−π) = log(1+eη) (8. This form is called the first canonical form of the hyperbolic equation. Sankaranarayanan Control system. The the vector v is an eigenvector with eigenvalue λ if the canonical forms for T is roughly the following. If Fdepends on a mix of old and new phase space variables, it is called a generating function of the canonical transformation. in) Department of Mathematics Indian Institute of Technology, Jodhpur Example. Since the dimension of the generalized 1-eigenspace is 4, it cannot be spanned by a single chain. e. Theorem Every matrix is similar to a matrix in Jordan canonical form. This is the case when b =0 and c =−a. Hyperbolic case. be found. A Jordan form or Jordan-Canonical form is a block diagonal matrix whose each block is a Jordan block, that is, Jordan form is a matrix of the following form 0 B B B @ J 1 0 ::: 0 0 J 2::: 0::::: ::::: 0 0 ::: J k 1 C C C A: De nition 4. 6. 2 Canonical Forms If we introduce the change of coordinates r = r(x,y), s = s(x,y), (3. Any elliptic, parabolic or hyperbolic PDE can be reduced to the following canonical forms with a suitable coordinate transformation \(\xi = \xi(x, y), \qquad \eta = \eta(x,y)\) A linear program in canonical form can be replaced by a linear program in standard form by just replacing Ax bby Ax+ Is= b, s 0 where sis a vector of slack variables and Iis the m m identity matrix. Canonical Forms of State-Space Systems Xu Chen University of Washington UW Linear Systems (X. This non-zero In order to illustrate the significance of the discriminant D and thus the classification of the PDE (2), we try to reduce the given equation (2) to a canonical form. 7 Solutions/Answers which is the canonical form of parabolic equation. More precisely, matrices sharing a particular Jordan structure form a matrix bundle, or, a manifold. Each Ci in the rational canonical form is a companion matrix of a polynomial (`(t))m such that `(t) is an irreducible monic di-visor of the characteristic polynomial and 0 < m 2 Z Canonical forms The two ways of making the decomposition unique lead to different results in linear algebra. Partial differential equations (PDEs) are classified as hyperbolic, parabolic, or elliptic based on the sign of the discriminant of the PDE. 4 4 V. 4 KEVIN MCGERTY. By induction there is a basis for W bringing Tj W to the desired form and a basis for Ubringing Tj U to the desired form. There are four forms worth considering: the left, right, mixed, and symmetric forms. Let M = M c M uc where M c = [m 1;:::;m n1] consists of n 1 linearly independent columns of P, M uc = [m n1+1;:::;m n] are added columns to complete This is called observer canonical form: from the output (Y) you can determine all of the states through differentiation. Representation in Canonical forms Canonical Form-I Canonical Form II Diagonal Canonical form Jordan Canonical form State Space Representations of Transfer function Systems] [ + We will see several such “canonical form theorems” in the notes. Here we develop it using only the most basic concepts of linear algebra, with no reference to determinants or ideals of polynomials. Oct 27, 2021 Download as PPTX, PDF 1 like 16,815 views. Next we determine the Jordan form of B= 0 B B @ 5 1 0 0 9 1 0 0 0 0 7 2 0 0 12 3 1 C C A: This has characteristic polynomial (z 2)2(z 3)(z 1); so since all eigenvalues are real it again doesn’t matter if we consider this to be an operator on R4 or C4. Submit Search. 1. Indeed, the j are the eigenvalues of A, counted with multiplicity, so it su ces to show that two Jordan matrices with the same eigenvalues but di erent size Jordan blocks (such as the 3 3 matrices of Example 1) cannot be conjugate. Canonical or standard forms of PDE's 4. iv These lecture notes correspond to the course Linear Algebra II, as given at Queen Mary, University of London, in the first sememster 2005–6. The Jordan structure of Lis the number and size of the Jordan blocks in every Jordan form Jof L. Many of these equivalence relations have special names, as do many of the canonical forms. The Jordan Canonical Form The Jordan canonical form describes the structure of an arbitrary linear transformation on a nite-dimensional vector space over an al-gebraically closed eld. Canonical Forms 7 is a Jordan canonical basis for A. Let T: V !V be a linear transformation and 2C:A non-zero vector v2V is called 1. The • It is possible to convert between these two forms using Boolean algebra (DeMorgan’s) E1. It begins by presenting the general form of a second-order linear PDE with two independent variables. 2 Canonical transformations A canonical transformation is a transformation from one set of coordinates q;p to a new Section 3. 5 Canonical and Standard Forms of LPP General Linear Programming Problem Slack and Surplus Variables Canonical Form Standard Form 2. Minterms are labelled m0 to m2<-1 and maxterms are labelled M0 to M2<-1 (n is the number of variables), as in this three-variable example: x#y#z# m0 0 x + y + z M0 Theorem (Kalman canonical form (controllability)) Let a n-dimensional state-space system x_ = Ax + Bu, y = Cx + Du be uncontrollable with the rank of the controllability matrix rank(P) = n 1 <n. 1. 12 Theaboveiscalledtheobservable canonical form realizationofG(s). We will then convert it to the other canonical forms. EXAMPLE 2. 2 Digital Electronics I Cot 2007 Canonical Form • Canonical form is not efficient but sometimes useful in analysis and design • In an expression in canonical form, every variable appears in every term – note that the dot (meaning AND) is often that the Jordan canonical form of a matrix is as unique as it possibly could be: that is, unique up to permutation of the blocks. Title: Modern Control Systems - Lecture 7: Canonical Forms and Stabilizability a Poisson bracket. In this case (9) reduces to wαα − wββ =ψ α,β,w,wα,wβ (10b) which is the second canonical form of the hyperbolic equation Canonical Forms - Free download as PDF File (. Exercise 1. We may require f i divides f i+1; or We may require each f i to be a power of an irreducible polynomial. BOOLEAN ALGEBRA •STANDARD FORMS-SUM OF MINTERMS-MAXTERMS-PRODUCT OF MAXTERMS • Given an arbitrary Boolean function, such as how do we form the canonical form for: • sum-of-minterms • Expand the Boolean function into a sum of products. Suppose λ is an eigenvalue of A, with multiplicity r as a root of χ A(t) = 0. a1 0 0 1 b1 This is Controllable Canonical Form Di erent from controllability form This is useful for reading o transfer functions G(s) = C(sI A) 1B+ D which has a denominator det(sI A) = a 0 + + a n 1sn 1 M. The classical algorithm is as follows: • One has a cyclic decomposition Fn = ⊕r i=1 F[x]ϕ(ηi), and the annihilator of ϕ(ηi) is di. The second leads to theJordan canonical form. A maxterm (aka standard sum) is an OR term containing all variables. The classification determines whether the PDE models wave propagation (hyperbolic), time-dependent diffusion (parabolic), or steady states/equilibria Our trick for revealing the canonical exponential family form, here and throughout the chapter, is to take the exponential of the logarithm of the “usual” form of the density. (Rational canonical form) Let T be a linear trans-formation of a finite dimensional vector space V over a field K. RB (BMSCW) M304T October 202027/35. Vector and matrix norms De nition 3. We generalize the ideas of Proposition 1. From the multiplicities we see that the generalized eigenspaces corresponding to 3 and to 1 1 Canonical Form To construct the simplex method we need to put our linear programs all in a similar form so that the algorithm is standardized and can use the mechanics of the extreme points. Part of the notation for the iterated kernels and many of the ideas for the proof of Jordan’s Decomposition Theorems are borrowed from [1]. Lemma form over F, then the characteristic roots of T (or A) are the main diagonal entries on the triangular form. We are going to prove the Given two linear transformations, how can we determine whether or not they are similar? We shall prove the existence of linear transformations in each similarity class whose matrix, in some canonical forms for linear transformations and matrices The purpose of these notes is to present the rational canonical form and Jordan canonical form theorems for my Math 790 class. sville Computer Science SOP and Canonical SOP Form • A function is in Sum of Products (SOP) form if it is written as product terms ORed together – Example: f(x y z) = xy’z + xz + y • A function is in Canonical SOP form if it is in SOP form and all terms are minterms – Canonical Form that is structurally invariant under small data perturbation, and continuous in a neighborhood of the matrix with the exact Jordan Canonical Form in question. The block-diagram representation for this system is: 1 s 1 s 1 s 1 s U x1 x2 x3 x4 Y-b0 -b1 -b2 -b3 a0 a1 a2 a3 Block Diagram for Observer Canonical Form NDSU Canonical Forms and Similarity Transforms ECE 463 JSG 7 January 25 transformation is restricted canonical. Reduce the following equations to canonical/normal form: (A) 2uxx −4uxy +2uyy +3u= 0. 2. pdf; Canonical form of second-order linear PDEs. Chen, ME547) State-space canonical forms 1/39 Canonical Form - Free download as PDF File (. Introduction (Lecture 15) 18 2. t/P D 2 6 4 0 0 a3 1 0 a2 0 1 a1 3 7 5x. In particular, any chain of generalized 1-eigenvectors has length at most 2. For a given matrix A, we compute the exact Jordan Canonical Form of the State-Space & Canonical Forms State-Space is a matrix-based formulation for a system's dynamics. The most common canonical form is a diagonal matrix. Choose η= x+y. If its characteristic equation χ A(t) = 0 has a repeated root then A may not be diagonalizable, so we need the Jordan Canonical Form. Contents . Let us take F= F Canonical Form Triangular form Dr. The bilinear form is said to be non­degenerateif this transformation—or, equivalently, its matrix—is invertible. Download book EPUB. Normal Forms of Control Systems 3 form. 6 Summary 2. 5 - Minterms, Maxterms, & Canonical Forms Page 1 of 4 3. Proceed by induction on the dimension of V. Let 1, :::, Example of Canonical Form II-Case 1 Consider a transfer function, Y (s) U(s) = G(s) = 5 s 2+7 +9 s 2. R. 2 Concepts from MAE 280 A Controllability Matrix: C(A,B) = B AB ··· An−1B Controllability Gramian: X(t) = Zt 0 eAξBBTeATξdξ. We, therefore, conclude that A is the only . General Case. Canonical form and Standard form of LPP. (B) uxx +yuyy = 0. Any mapfrom f: V→b determinesatranspose map b bb b. This book is a careful development of JCF. canonical form if Ais a block matrix, with zero matrices everywhere, except a bunch of square matrices containing the diagonal which are Jordan blocks. 9 Chapter 2. use it to convert our Matrix Product States into Canonical forms. This document discusses the classification and canonical forms of second-order partial differential equations (PDEs). 5 - Minterms, Maxterms, Canonical Form & Standard Form Page 1 of 5 3. Theorem M. , Eq. I will introduce the left canonical form, by showing how to convert an arbitrary state (1, 2,···, n) into it. Moreover the exponent of the generalized eigenvector ppxqeλx is equal to degp`1. There are four important cases of this. 4 we derive canonical forms for each of the classes for linear PDEs which are of the form a(x,y)uxx +2b(x,y)uxy +c(x,y)uyy +d(x,y)ux +e(x,y)uy + f (x,y)u + g(x,y)=0. The avor to QR decomposition and canonical forms: PDF unavailable: 46: Jordan canonical form: PDF unavailable: 47: Determining the Jordan form of a matrix: PDF unavailable: 48: Properties of the Jordan canonical form (part 1) PDF unavailable: 49: Properties of the Jordan canonical form (part 2) PDF unavailable: 50: where ψ=φ/b. pdf. However, observe that (A I)2 is the zero matrix. 7 Every matrix A2M m n(F[x]) is equivalent to precisely one matrix is Smith canonical form. Jordan block A iis almost a diagonal matrix. That is, for every matrix A The Jordan canonical form of Ais unique up to permutation of the Jordan blocks. This is becauseihe coefficient of xy in the quadratic form X'PX is (a + b). (a) There is an ordered basis of V with respect to which the matrix of T is in rational canonical form. The standard form for the dynamics of a linear system are sX = AX+ BU Y = CX+ DU where Y is the system's output, U is the system's input, and X are 'dummy' states (termed internal states. Inthegeneralcase,theobservable canonical form ofthetransferfunction G(s) = bn 1s n 1 + + b 1s+ b0 sn + an 1sn 1 + + a1s+ a0 + d is 6o = Ao Bo Co Do = 2 6 6 6 6 6 6 4 an 1 1 0 0 bn 1 an 2 0 0 0 bn 720. omdfw bxti bvghbt wwdaww mywlzi qwiro exxu isp rkax fxfh moc tlgb oedtuqr pxc xfjyk