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Capitolo 0. INTRODUCTION 2.1 Diagonalizable matrices If matrix A has n real and distinct eigenvalues λi, then it also has n real • eigenvectors vi which are linearly independent: Avi = λi vi, i 1, 2, 3, ..., n . ∈ { } In this case the matrix A can be diagonalized using a transformation matrix T having the eigenvectors vi as its columns: T = v1 v2 v3 ... vn 1 The transformed matrix A = T− AT is diagonal. The coefficients on the • diagonal of matrix A are equal to the eigenvalues λi of matrix A: λ1 λ2 1 A = T− AT = λ3 .. . λn The order of the eigenvalues λi on the diagonal of matrix A if equal to the • order of the the corresponding eigenvectors vi within matrix T. A matrix A can be diagonalized if and only if it is possible to find a number • of linear independent eigenvalues vi equal to the dimension n of matrix A. The matrices which are not diagonalizable can be transformed in the Jordan • canonical form. This canonical form is the form more likely diagonal. A matrix A can be diagonalized also if its eigenvalues λi are complex • conjugate. In this case the eigenvectors vi are complex conjugate and the transformation T is complex. Zanasi Roberto - System Theory. A.A. 2015/2016 Capitolo 2. CANONICAL FORMS 2.2 Jordan canonical form Let λi, for i =1, ..., h, be the “distinct” eigenvalues of matrix A and let ri • be the corresponding molteplicity degree within the characteristic polynomial: r1 r2 rh ∆A(λ) = (λ λ ) (λ λ ) ... (λ λh) − 1 − 2 − A matrix A transformed in the Jordan canonical form A has the following • block diagonal form: J1 0 ... 0 0 J ... 0 A = T 1AT = 2 − . 0 0 ... Jh To each distinct eigenvalue λi of matrix A corresponds Jordan block Ji of • dimension equal to the algebraic molteplicity ri of eigenvalue λi, that is the molteplicity ri of eigenvalue λi within the characteristic polynomial ∆A(λ). Each Jordan block Ji has itself the structure of a block diagonal matrix: • J 0 ... 0 i,1 dim J = r 0 J ... 0 i i J = i,2 i . i =1, ..., h 0 0 ... Ji,mi On the diagonal of the Jordan block Ji are present mi Jordan miniblocks • Ji, j, where mi is the geometric molteplicity of the eigenvaalue λi, that is the number of linear independent eigenvectors vi,j associated to the eigenvalue λi. The structure of all the Jordan miniblocks Ji, j is the following: • λi 1 0 ... 0 0 0 λi 1 ... 0 0 dim Ji,j = νi,j 0 0 λi ... 0 0 Ji,j = . . j =1, ..., mi 0 0 0 ... λi 1 0 0 0 ... 0 λi Zanasi Roberto - System Theory. A.A. 2015/2016 Capitolo 2. CANONICAL FORMS 2.3 The following relations hold: • h mi n = ri, ri = νi,j. i=1 j=1 X X The matrix A is diagonalizable if and only if the dimension νi,j of all the • Jordan miniblocks Ji, j is unitary: νi,j =1. In this case all the Jordan miniblocks Ji, j = λi are equal to the corresponding • eigenvalue λi and all the Jordan blocks Ji are diagonal and characterized by the same eigenvalue λi: λ 0 ... 0 i dim J = r 0 λ ... 0 i i J = λ , J = i , i, j i i . i =1, ..., h 0 0 ... λi A matrix A is diagonalizable if and only if the algebraic molteplicity ri • of each eigenvalue λi is equal to the geometric molteplicity mi, that is if the number mi of linearly independent eigenvectors vi,j associated to each eigenvalue λi is equal to the algebraic molteplicity ri of the eigenvalue λi within the characteristic polynomial ∆A(λ). Example. Matrix in Jordan canonical form: • 2 3 1 1 ∆A(λ)=(λ + 1) (λ + 3) − 0 1 J1,1 n =5, h =2 − J1 A= 3 = J2,1 = , λ1 = 1, λ2 = 3 − J2 − − 3 1 J2,2 r =2, r =3 − 1 2 0 3 m1 =1, m2 =2 − The matrix A has two distinct eigenvalues λ = 1 and λ = 3. The eigenvalue λ has 1 − 2 − 1 algebraic molteplicity r1 = 2 and geometric molteplicity m1 = 1. The eigenvalue λ2 has algebraic molteplicity r2 =3 and geometric molteplicity m2 =2. The Jordan block J1 has dimension r1 =2 and it is composed by only one miniblock J1,1. The second Jordan block J2 has dimension r2 =3 and it is composed by m2 =2 miniblocks J2,1 and J2,2. Zanasi Roberto - System Theory. A.A. 2015/2016 Capitolo 2. CANONICAL FORMS 2.4 The mi linear independent eigenvectors vi,j associated to the eigenvalue λi • can be determined solving the following autonomous linear system: (λiI A)vi,j =0, j =1, ..., mi. − The number mi of linearly independent eigenvectors vi,j is equal to the • number of miniblocks Ji,j present within the Jordan block Ji. In the case mi < ri, the number of linearly independent eigenvectors vi,j • (k) is not sufficient for diagonalizing the matrix. In this case, the chains vi,j of generalized eigenvectors for i = 1, 2, ..., h, j = 1, 2, ..., mi and k =1, 2, ..., νi,j must be determined. The generalized eigenvectors v(k) can be determined solving the following • i,j linear equations: (2) (1) (A λiI)v = v = vi,j − i,j i,j (3) (2) (A λiI)v = v − i,j i,j . (νi,j) (νi,j 1) (A λiI)vi,j = vi,j − − (1) The first eigenvectorvi,j = vi,j is known. Solving the first equation one (2) (2) obtains the generalized eigenvector vi,j . Substituting vi,j and solving the (3) next equation one obtains the generalized eigenvector vi,j , ..., and so on. The particular “almost diagonal” form of the Jordan miniblocks Ji,j is ob- • (k) tained inserting the chains of generalized eigenvectors vi,j as columns of the transformation matrix T: (1) (2) (νi,j) T = ... vi,j vi,j ... vi,j ... h i If the geometric molteplicity mi is equal to the algebraic molteplicity ri, that • (k) is if mi = ri, the chain of generalized eigenvectors vi,j has unitary length and it is formed by only one eigenvector vi,j. Zanasi Roberto - System Theory. A.A. 2015/2016 Capitolo 2. CANONICAL FORMS 2.5 Numeric example in Matlab: • --------------------------------------------------------------------------- clc; echo on %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrix to be transformed in Jordan canonical form %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A =sym([... 73/129, -18/43, 1/129, 383/129, -1213/258; 2610/989, -2419/989, -687/989, 1759/989, -6889/1978; 2885/989, 655/989, -4039/989, 2310/989, -3905/989; 2273/989, -2079/989, 1565/989, -326/989, -5915/1978; 4456/2967, -3052/989, 9190/2967, 4778/2967, -13964/2967]); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The command "jordan(A)" transforms matrix A in Jordan canonical form %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [V,AJ]=jordan(A); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AJ %%% Jordan canonical form of marix A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AJ = [ -3, 1, 0, 0, 0] [ 0, -3, 0, 0, 0] [ 0, 0, -1, 1, 0] [ 0, 0, 0, -1, 0] [ 0, 0, 0, 0, -3] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V %%% Matrix of generalized eigenvectors of matrix A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V = [ -522/989, -242/989, 4760/2967, 1231/989, 6450201/4106690] [ -406/989, -3334/2967, 2380/2967, 3334/2967, -142348/293335] [ -290/989, -3275/2967, 2975/2967, 3275/2967, -903187/1642676] [ -348/989, -418/989, 1785/989, 418/989, -186498/2053345] [ -580/989, -718/2967, 4760/2967, 718/2967, 966065/821338] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Verification: the columns of V are the generalized eigenvectors of A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (A-(-3)*eye(5))*V(:,1)==0 % --> Vero (A-(-3)*eye(5))*V(:,2)==V(:,1) % --> Vero (A-(-1)*eye(5))*V(:,3)==0 % --> Vero (A-(-1)*eye(5))*V(:,4)==V(:,3) % --> Vero (A-(-3)*eye(5))*V(:,5)==0 % --> Vero --------------------------------------------------------------------------- Zanasi Roberto - System Theory. A.A. 2015/2016 Capitolo 2. CANONICAL FORMS 2.6 The free evolution of a discrete linear system is the following: • k 1 k k 1 x(k) = A x0 = (TAT− ) x0 = TA T− x0 k k J1 0 ... 0 J1 0 ... 0 0 J ... 0 0 Jk ... 0 = T 2 T 1x = T 2 T 1x . − 0 . − 0 k 0 0 ... Jh 0 0 ... Jh The free evolution of a time-continuous linear system is the following: • At (TAT 1)t At 1 x(t) = e x0 = e − x0 = Te T− x0 J1 0 ... 0 0 J2 ... 0 J1t . t e 0 ... 0 . J2t 0 0 ... Jh 1 0 e ... 0 1 = T e T− x0 = T . T− x0 . J t 0 0 ... e h The power Ak and the exponential eAt of matrix A can be determined if it • k Jit is known how to compute the power Ji and the exponential e of the generic Jordan miniblock Ji of dimension ν: λ 1 0 ... 0 0 0 λ 1 ... 0 0 0 0 λ ... 0 0 J = . = λI + N . 0 0 0 ... λ 1 0 0 0 ... 0 λ Matrix J can be expressed as a sum of the diagonal matrix λI and the • nilpotent matrix N which has non zero and unitary elements only on the first over-diagonal. In the case ν =5, it is: 01000 00100 N = 00010 00001 00000 Zanasi Roberto - System Theory. A.A. 2015/2016 Capitolo 2. CANONICAL FORMS 2.7 The powers of matrix N have the following structure: • 00100 00010 00010 00001 N2 = 00001 N3 = 00000 ... 00000 00000 00000 00000 that is, the matrix Nk has non zero elements only on the k-th over-diagonal. The matrix N is nilpotent of order ν if it satisfies the following relation: • Nν =0 dove ν = dim N. The k-th power of matrix J can be expressed in the following way: • k k Jk = (λI + N)k = λkI + λk 1N + λk 2N2 + ..
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