Lecture 12 Jordan canonical form

1 EE263 Autumn 2007-08 Stephen BoydLecture 12 Jordan canonical form Jordan canonical form generalized modes Cayley-Hamilton theorem12 1 Jordan canonical formwhat ifAcannot be diagonalized?anymatrixA Rn ncan be put inJordan canonical formby a similaritytransformation, 1AT=J= whereJi= i1 i Cni niis called aJordan blockof sizeniwith eigenvalue i(son=Pqi=1ni) Jordan canonical form12 2 Jis upper bidiagonal Jdiagonal is the special case ofnJordan blocks of sizeni= 1 Jordan form is unique (up to permutations of the blocks) can have multiple blocks with same eigenvalueJordan canonical form12 3note:JCF is aconceptual tool, never used in numerical computations!X(s) = det(sI A) = (s 1)n1 (s q)nqhence distinct eigenvalues ni= 1 AdiagonalizabledimN( I A)is the number of Jordan blocks with eigenvalue more generally,dimN( I A)k=X i= min{k, ni}so fromdimN( I A)kfork= 1,2.

2 We can determine the sizes ofthe Jordan blocks associated with Jordan canonical form12 4 factor outTandT 1, I A=T( I J)T 1 for, say, a block of size3: iI Ji= 0 1 00 0 10 0 0 ( iI Ji)2= 0 0 10 0 00 0 0 ( iI Ji)3= 0 for other blocks (say, size 3, fork 2)( iI Jj)k= ( i j)k k( i j)k 1(k(k 1)/2)( i j)k 20( j i)k k( j i)k 100( j i)k Jordan canonical form12 5 Generalized eigenvectorssupposeT 1AT=J=diag(J1, .. , Jq)expressTasT= [T1T2 Tq]whereTi Cn niare the columns ofTassociated withith Jordan blockJiwe haveATi=TiJiletTi= [vi1vi2 vini]then we have:Avi1= ivi1, , the first column of eachTiis an eigenvector associated with iforj= 2, .. , ni,Avij=vi j 1+ ivijthe vectorsvi1, .. viniare sometimes calledgeneralized eigenvectorsJordan canonical form12 6 Jordan form LDSconsider LDS x=Axby change of coordinatesx=T x, can put into form x=J xsystem is decomposed into independent Jordan block systems xi=Ji xi x1 xni xni 11/s1/s1/s Jordan blocks are sometimes called Jordan chains(block diagram shows why) Jordan canonical form12 7 Resolvent, exponential of Jordan blockresolvent ofk kJordan block with eigenvalue :(sI J ) 1= s 1s.

3 1s 1= (s ) 1(s ) 2 (s ) k(s ) 1 (s ) k+ (s ) 1 = (s ) 1I+ (s ) 2F1+ + (s ) kFk 1whereFiis the matrix with ones on theith upper diagonalJordan canonical form12 8by inverse Laplace transform, exponential is:etJ =et I+tF1+ + (tk 1/(k 1)!)Fk 1 =et 1t tk 1/(k 1)!1 tk 2/(k 2)!..1 Jordan blocks yield: repeated poles in resolvent terms of formtpet inetAJordan canonical form12 9 Generalized modesconsider x=Ax, withx(0) =a1vi1+ +anivini=Tiathenx(t) =T eJt x(0) =TieJita trajectory stays in span of generalized eigenvectors coefficients have formp(t)e t, wherepis polynomial such solutions are calledgeneralized modesof the systemJordan canonical form12 10with generalx(0)we can writex(t) =etAx(0) =T etJT 1x(0) =qXi=1 TietJi(STix(0))whereT 1= hence: all solutions of x=Axare linear combinations of (generalized)modesJordan canonical form12 11 Cayley-Hamilton theoremifp(s) =a0+a1s+ +akskis a polynomial andA Rn n, we definep(A) =a0I+a1A+ +akAkCayley-Hamilton theorem.

4 For anyA Rn nwe haveX(A) = 0, whereX(s) = det(sI A)example:withA= 1 23 4 we haveX(s) =s2 5s 2, soX(A) =A2 5A 2I= 7 1015 22 5 1 23 4 2I= 0 Jordan canonical form12 12corollary:for everyp Z+, we haveAp span I, A, A2, .. , An 1 (and ifAis invertible, also forp Z) , every power ofAcan be expressed as linear combination ofI, A, .. , An 1proof:divideX(s)intospto getsp=q(s)X(s) +r(s)r= 0+ 1s+ + n 1sn 1is remainder polynomialthenAp=q(A)X(A) +r(A) =r(A) = 0I+ 1A+ + n 1An 1 Jordan canonical form12 13forp= 1: rewrite C-H theoremX(A) =An+an 1An 1+ +a0I= 0asI=A (a1/a0)I (a2/a0)A (1/a0)An 1 (Ais invertible a06= 0) soA 1= (a1/a0)I (a2/a0)A (1/a0)An , inverse is linear combination ofAk,k= 0, .. , n 1 Jordan canonical form12 14 Proof of C-H theoremfirst assumeAis diagonalizable:T 1AT= X(s) = (s 1) (s n)sinceX(A) =X(T T 1) =TX( )T 1it suffices to showX( ) = 0X( ) = ( 1I) ( nI)=diag(0, 2 1.)

5 , n 1) diag( 1 n, .. , n 1 n,0)= 0 Jordan canonical form12 15now let s do general case:T 1AT=JX(s) = (s 1)n1 (s q)nqsuffices to showX(Ji) = 0X(Ji) = (Ji 1I)n1 0 1 0 0 0 1 .. ni|{z}(Ji iI)ni (Ji qI)nq= 0 Jordan canonical form12 16