18.700 JORDAN NORMAL FORM NOTES These are some supplementary notes on how to find the Jordan normal form of a small matrix. First we recall some of the facts f rom lecture, next we give the general algorithm for finding the Jordan normal form of a linear operator, and then we will see how this works for small matrices. 1. Facts Throughout we will work over the field C of complex numbers, but if you like you may replace this with any other algebraically closed field. Suppose that V is a C-vector space of dimension n and suppose that T : V V is a C-linear operator. Then the characteristic →polynomial of T factors into a product of linear terms, and the irreducible factorization has the form cT (X) = (X − λ1)m1 (X − λ2)m2 . . . (X − λr )mr , (1) for some distinct numbers λ1, . . . , λr ∈ C and with each mi an integer m1 ≥ 1 such that + mr = n.m1 + · · ·Recall that for each eigenvalue λi, the eigenspace Eλi is the kernel of T − λiIV . We generalized this by defining for each integer k = 1, 2, . . . the vector subspace E(X−λi )k = ker(T − λiIV )k . (2) It is clear that we have inclusions Eλi = EX−λi ⊂ E(X−λi)2 ⊂ . . . . (3) ⊂ · · · ⊂ E(X−λi)e Since dim(V ) = n, it cannot happen that each dim(E(X−λi)k ) < dim(E(X−λi)k+1 ), for each k = 1, . . . , n. Therefore there is some least integer ei ≤ n such that E(X−λi )ei = E(X−λi)ei+1 . As was proved in class, for each k ≥ ei we have E(X −λi)k = E(X−λi)ei , and we defined the generalized eigenspace Egen to be E(X−λi)ei .λi It was proved in lecture that the subspaces Egen, . . . , Egen give a direct sum decomposition λ1 λr of V . From this our criterion for diagonalizability of follows: T is diagonalizable iff for each i = 1, . . . , r, we have Egen = Eλi . Notice that in this case T acts on each Egen as λi timesλi λi the identity. This motivates the definition of the semisimple part of T as the unique C-linear operator S : V → V such that for each i = 1, . . . , r and for each v ∈ Egen we have S(v) = λiv.λi We defined N = T − S and observed that N preserves each Egen and is nilpotent, i.e. there λi exists an integer e ≥ 1 (really just the maximum of e1, . . . , er ) such that Ne is the zero linear operator. To summarize: (A) The generalized eigenspaces Egen , . . . , Egen defined by λ1 λr Egen = {v ∈ V |∃e, (T − λiIV )e(v) = 0}, (4) λi Date: Fall 2001. 1� � � �� � � � � � � � 2 18.700 JORDAN NORMAL FORM NOTES give a direct sum decomposition of V . Moreover, we have dim(Egen) equals the algebraic λi multiplicity of λi, mi. (B) The semisimple part S of T and the nilpotent part N of T defined to be the unique C-linear operators V → V such that for each i = 1, . . . , r and each v ∈ Egen we have λi S(v) = S(i)(v) = λiv, N(v) = N(i)(v) = T (v) − λiv, (5) satisfy the properties: (1) S is diagonalizable with cS (X) = cT (X), and the λi-eigenspace of S is Egen (for T ).λi (2) N is nilpotent, N preserves each Egen and if N(i) : Egen λi is the unique linear λi λi Egen →ei−1 eiN(i)operator with N(i)(v) = N(v), then N(i)is nonzero but = 0. (3) T = S + N . (4) SN = NS. (5) For any other C-linear operator T� : V V , T� commutes with T (T�T = T T�) iff T�→commutes with both S and N . Moreover T� commutes with S iff for each i = 1, . . . , r, (Egen) ⊂ Egen we have T�λi λi . (6) If (S�, N�) is any pair of a diagonalizable operator S� and a nilpotent operator N� such that T = S� + N� and S�N� = N�S�, then S� = S and N� = N. We call the unique pair (S, N) the semisimple-nilpotent decomposition of T . (i)(C) For each i = 1, . . . , r, choose an ordered basis B(i) = (v(i) mi ) of Egen and let , . . . , v1 λi B(1), . . . , B(r) be the concatenation, i.e. B = (1) (1) (2) (2) (r)B = v1 , . . . , vm1 , v1 , . . . , vm2 , . . . , v1 , . . . , v(r) . (6) mr For each i let S(i), N(i) be as above and define the mi × mi matrices N(i)D(i) = S(i) B(i),B(i) , C(i) = B(i),B(i) . (7) Then we have D(i) = λiImi and C(i) is a nilpotent matrix of exponent ei. Moreover we have the block forms of S and N: ⎞⎛ λ1Im1 0m1×m2 . . . 0m1×mr 0m2×m1 λ2Im2 . . . 0m2×mr ⎜⎜⎝ ⎟⎟⎠ [S](8) = ,. ...B,B . . . ... . . 0mr ×m1 0mr ×m1 . . . λr Imr C(1) 0m1×m2 . . . 0m1×mr 0m2×m1 C(2) . . . 0m2×mr ⎞⎛ ⎜⎜⎜⎝ ⎟⎟⎟⎠ [N](9) = .. ...B,B . . . ... . . C(r). . . 0mr ×m1 0mr ×m2 Notice that D(i) has a nice form with respect to ANY basis B(i) for Egen . But we might hope λi to improve C(i) by choosing a better basis.� � � � 3 18.700 JORDAN NORMAL FORM NOTES A very simple kind of nilpotent linear transformation is the nilpotent Jordan block, i.e. TJa : Ca Ca where Ja is the matrix → ⎞⎛ 0 0 0 . . . 0 0 1 0 0 . . . 0 0 0 1 0 . . . 0 0 ⎜⎜⎜⎜⎜⎜⎝ ⎟⎟⎟⎟⎟⎟⎠ J. (10) = . . .. . .a . . . . . .... . . . 0 0 0 . . . 0 0 0 0 0 . . . 1 0 ⎞ In other words, Jae1 = e2, Jae2 = e3, . . . , Jaea−1 = ea, Jaea = 0. (11) Notice that the powers of Ja are very easy to compute. In fact Ja = 0a,a, and for d = a 1, . . . , a − 1, we have Jad e1 = ed+1, Jad e2 = ed+2, . . . , Jad ea−d = ea, Jd ea+1−d = 0, . . . , Jd ea = 0. (12) a a Notice that we have ke r(Jad) = span(ea+1−d, ea+2−d, . . . , ea). A nilpotent matrix C ∈ Mm×m(C) is said to be in Jordan normal form if it is of the form …
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