DOC PREVIEW
MIT 18 034 - JORDAN NORMAL FORM NOTES

This preview shows page 1-2-3-4 out of 12 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

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 …


View Full Document

MIT 18 034 - JORDAN NORMAL FORM NOTES

Documents in this Course
Exam 3

Exam 3

9 pages

Exam 1

Exam 1

6 pages

Load more
Download JORDAN NORMAL FORM NOTES
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view JORDAN NORMAL FORM NOTES and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view JORDAN NORMAL FORM NOTES 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?