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Chapter 6 Jordan Canonical Form

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Chapter 6 Jordan Canonical Form Chapter 6 Jordan Canonical Form 175 6.1 Primary Decomposition Theorem For a general matrix, we shall now try to obtain the analogues of the ideas we have in the case of diagonalizable matrices. Diagonalizable Case General Case Minimal Polynomial: Minimal Polynomial: k k Y rj Y rj mA(λ) = (λ − λj) mA(λ) = (λ − λj) j=1 j=1 where rj = 1 for all j where 1 ≤ rj ≤ aj for all j Eigenspaces: Eigenspaces: Wj = Null Space of (A − λjI) Wj = Null Space of (A − λjI) dim(Wj) = gj and gj = aj for all j dim(Wj) = gj and gj ≤ aj for all j Generalized Eigenspaces: rj Vj = Null Space of (A − λjI) dim(Vj) = aj Decomposition: Decomposition: n n F = W1 ⊕ W2 ⊕ Wj ⊕ · · · ⊕ Wk W = W1⊕W2⊕Wj⊕· · ·⊕Wk is a subspace of F Primary Decomposition Theorem: n F = V1 ⊕ V2 ⊕ Vj ⊕ Vk n n x 2 F =) 9 unique x 2 F =) 9 unique xj 2 Wj, 1 ≤ j ≤ k such that vj 2 Vj, 1 ≤ j ≤ k such that x = x1 + x2 + ··· + xj + ··· + xk x = v1 + v2 + ··· + vj + ··· + vk 176 6.2 Analogues of Lagrange Polynomial Diagonalizable Case General Case Define Define k k Y Y ri fj(λ) = (λ − λi) Fj(λ) = (λ − λi) i=1 i=1 i6=j i6=j This is a set of coprime polynomials This is a set of coprime polynomials and and hence there exist polynomials hence there exist polynomials Gj(λ), 1 ≤ gj(λ), 1 ≤ j ≤ k such that j ≤ k such that k k X X fj(λ)gj(λ) = 1 Fj(λ)Gj(λ) = 1 j=1 j=1 The polynomials The polynomials `j(λ) = fj(λ)gj(λ); 1 ≤ j ≤ k Lj(λ) = Fj(λ)Gj(λ); 1 ≤ j ≤ k are the Lagrange Interpolation are the counterparts of the polynomials Lagrange Interpolation polynomials `1(λ) + `2(λ) + ··· + `k(λ) = 1 L1(λ) + L2(λ) + ··· + Lk(λ) = 1 λ1`1(λ)+λ2`2(λ)+···+λk`k(λ) = λ λ1L1(λ) + λ2L2(λ) + ··· + λkLk(λ) = d(λ) (a polynomial but it need not be = λ) For i 6= j the product `i(λ)`j(λ) has For i 6= j the product Li(λ)Lj(λ) has a a factor mA (λ) factor mA (λ) 6.3 Matrix Decomposition In the diagonalizable case, using the Lagrange polynomials we obtained a decomposition of the matrix A. We shall next look at the analogous situation for a general matrix. 177 Diagonalizable Case General Case Define the matrices Analogously define the matrices Aj = `j(A) for 1 ≤ j ≤ k Aj = Lj(A) for 1 ≤ j ≤ k We have Analogously we have A1 + A2 + ··· + Ak = I A1 + A2 + ··· + Ak = I since since `1(λ) + `2(λ) + ··· + `k(λ) = 1 L1(λ) + L2(λ) + ··· + Lk(λ) = 1 AiAj = 0(n×n) if i 6= j AiAj = 0(n×n) since, then `i(λ)`j(λ) has the mini- since, then Li(λ)Lj(λ) has the minimal mal polynomial as a factor polynomial as a factor 2 2 Aj = Aj for 1 ≤ j ≤ k Aj = Aj for 1 ≤ j ≤ k Range of Aj is Wj and Ajx = x for Range of Aj is Vj and Ajx = x for all all x 2 Wj x 2 Vj We have We will not have λ1A1 + λ2A2 + ··· + λkAk = A λ1A1 + λ2A2 + ··· + λkAk = A since since λ1`1(λ)+λ2`2(λ)+···+λk`k(λ) = λ λ1L1(λ) + λ2L2(λ) + ··· + Lk(λ) ; in general, may not be equal to λ. Let D = λ1A1 + λ2A2 + ··· + λkAk Then D is a diagonalizable matrix (by The- orem 5.7.1). 178 We then look at by how much A differs form D. For this purpose we define N = A − D so that A = D + N (In the case where A is a diagonalizable matrix we have N = 0(n×n) and A = D). Thus, in the general case, in addition to a diagonalizable matrix, we also have to analyse the part N = A − D. We shall do this in the next section. 6.4 Nilpotent Matrices We want to analyse the difference N = A − D (6.4.1) where D is as defined in the previous section. Using the fact that I = A1 + A2 + ··· + Ak we get A = AIn×n = AA1 + AA2 + ··· + AAk Combining this and the fact that D = λ1A1 + λ2A2 + ··· + λkAk we get A − D = (AA1 + AA2 + ··· + AAk) − (λ1A1 + λ2A2 + ··· + λkAk) = (A − λ1In×n)A1 + (A − λ2In×n)A2 + ··· + (A − λkIn×n)Ak All the Aj, being polynomials in A, commute with A, and furher, AiAj = 0(n×n). Using these facts we get r r r r (A − D) = (A − λ1In×n) A1 + (A − λ2In×n) A2 + ··· + (A − λkIn×n) Ak (6.4.2) r If r ≥ Max: fr1; r2; ··· ; rjg then (A − λjIn×n) Aj has a factor mA (A), and hence r (A − λjIn×n) Aj = 0n×n for all j if r ≥ Max:fr1; r2; ··· ; rjg (6.4.3) 179 Hence we get from (6.4.2) r r N = (A − D) = 0n×n for r ≥ Max:fr1; r2; ··· ; rjg (6.4.4) Thus the difference between A and the diagonalizable matrix D is such that a power of it vanishes. This leads us to the notion of a nilpotent matrix. We have n×n Definition 6.4.1 A matrix N 2 C is said to be a \NILPOTENT" r matrix if there exists a positive integer r such that N = 0n×n. For r a nilpotent matrix the smallest power r for which N = 0n×n is called the \order of nilpotency" of N and is denoted by γ . N Thus we have that the matrix N = A − D is a nilpotent matrix the order of nilpotency being equal to Max: fr1; r2; ··· ; rkg. We can summarise our analysis above as follows: n×n Theorem 6.4.1 Every matrix A 2 C can be expressed as the sum, A = D + N; (6.4.5) of a diagonalizable matrix D and a nilpotent matrix N We have already analysed diagonalizable matrices in the last chapter. We shall now analyse a nilpotent matrix. 6.5 Nilpotent Matrices Let N 2 n×n be a nilpotent matrix with order of nilpotency γ . This means F N that γ N N = 0n×n and (6.5.1) N r 6= 0 for any r < γ (6.5.2) n×n N From (6.5.1) we also get, N r = 0 for all r ≥ γ (6.5.3) n×n N γ From this it follows that the polynomial p(λ) = λ N is an annihilating poly- γ nomial for N. Hence the minimal polynomial must divide λ N . Thus the 180 minimal polynomial must be of the form m (λ) = λr where r ≤ γ . How- N N ever, from (6.5.2) we have that N r 6= 0 for any r < γ . Hence we get n×n N that the minimal polynomial of N is given by γ N mN (λ) = (λ) Thus we see that λ1 = 0 is the only eigenvalue of any nilpotent matrix N. Since the characteristic polynomial of N is a monic polynomial of degree n and its roots are the eigenvalues we get that the characteristic polynomial of N is given by n cN (λ) = λ Hence we have, n×n Theorem 6.5.1 If N 2 F is a nilpotent matrix with order of nilpotency as γ then its characteristic and minimal polynomials N are given by n cN (λ) = λ , and (6.5.4) γ N mN (λ) = λ (6.5.5) From the above simple properties we now obtain a useful result. Suppose A 2 n×n is both nilpotent and diagonalizable. Since it is nilpotent matrix F γ A its minimal polynomial is of the form mA = λ , where γA is its order of nilpotency. Combining this with the fact that A is diagonalizable we get that its minimal polynomial must be mA (λ) = λ. Hence mA (A) = 0n×n gives us that A = 0n×n. Thus it follows that n×n Theorem 6.5.2 The only matrix in F which is both diagonaliz- able and nilpotent is the zero matrix We shall now see an example to show that the sum and product of two nilpotent matrices need not be nilpotent. n×n Example 6.5.1 . Consider the matrices N1;N2 2 C given below: 0 1 ! N = (6.5.6) 1 0 0 0 0 ! N = (6.5.7) 2 1 0 181 It is easy to see that 2 N1 = 02×2 2 N2 = 02×2 Thus both are nilpotent matrices both with order of nilpotency as 2. How- ever, we have, 0 1 ! N + N = 1 2 1 0 This matrix has characteristic polynomial given by λ2 − 1 = 0 Since there are two eigenvalues λ1 = 1; λ2 = −1, both eigenvalues having algebraic and geometric multiplicity both as 1, the matrix is diagonalizable. Since this is not the zero matrix, by the Theorem 6.5.2, this matrix cannot be nilpotent. Thus we see that the sum N1 +N2 of the two nilpotent matrices N1 and N2 is not nilpotent. Similarly we have, 1 0 ! N N = 1 2 0 0 and this is a diagonal matrix. Again by Theorem 6.5.2 it follows that N1N2 is not nilpotent. Thus the product N1N2 of the two nilpotent matrices N1 and N2 is not nilpotent. However, we can show that the following is true: when two nilpotent matrices commute with each other, then their sum and product will also be nilpotent We shall now use this result to investigate the uniqueness of the decompo- sition of a matrix as the sum D + N of a diagonalizable matrix D and a nilpotent matrix N obtained in Section 6.3.
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