2 FALL 2017 (c)Forλ2=3 rank(A-3I)=rank(J-31)=7, rank(A-31)2 rank(J3)2=4, rank(A-31)rank(J-3)3=3. rank(A-31)=rank(-31)=2. rank(A-31)rank()=2.k 24 12 7-=3 is the number of blocks with dia al a with dimer 4-3=1 is the number of blocks with diagonal 3 with dimension 3x3. gonal 3 with dimension 4x 4. (d)Suppose S (Ao7 =307 Ao3=303 Aas =305 Aa2=2a2+a4 Aa4=304+03 a6=306十a5 4a10=3a10+a8 There are 4 eigenvectors Notice that 4 is the number of blocks in the Jordan form 1.2.Properties.Jordan form is the standard form or simple representative in each class given by similar equivalence. AS=SJ.J=diag(J()..J(e)),J()=ding ()...() 入九1 ()= (a)First,J is block diagonal with each diagonal block (co sponding to oneλ.Second.each ()is block diagonal such that the small diagonal blocks all have on the diagonal,1 upper next to diagona and 0 at o of small blockssth multiplicity nCorresponding to eachthe number (b)The number and dimensions for those small blocks corresponding to each is fully determined by rank(A-)'=r,s=1,2, n-r1 =number of blocks corresponding to Ai with dimension 2 1 x 1 r-ra=number of blocks corresponding to with dimension2x2 ra-ra=number of blocks corresponding to with dimension3 (n-r)-(r-ra)=number of blocks corresponding to with dimension 1x1 (r1-r2)-(r2-rs)=number of blocks corresponding to A;with dimension 2 x 2 first si such that r)Then corresponding to this,the (c)Jordan form determines all the pr operties of A that does not change under similar transformation. For example,eigenvalues,characte ristic polynomial,minimal polynomial. Then those colmn vectors,which corresponds to the first term of each smal2 FALL 2017 (c) For λ2 = 3 rank(A − 3I) = rank(J − 3I) = 7, rank(A − 3I) 2 = rank(J − 3I) 2 = 4, rank(A − 3I) 3 = rank(J − 3I) 3 = 3, rank(A − 3I) 4 = rank(J − 3I) 3 = 2, rank(A − 3I) k = rank(J − 3I) k = 2, ∀k ≥ 4 4 is the largest block dimension with diagonal 3. 10 − 7 = 3 is the number of blocks with diagonal 3. 7 − 4 = 3 is the number of blocks with diagonal 3 with dimension ≥ 2 × 2. 4 − 3 = 1 is the number of blocks with diagonal 3 with dimension ≥ 3 × 3. 3 − 2 = 1 is the number of blocks with diagonal 3 with dimension ≥ 4 × 4. (d) Suppose S = [α1, ..., α10] such that AS = SJ. Then ( Aα1 = 2α1 Aα2 = 2α2 + α1 ( Aα3 = 3α3 Aα4 = 3α4 + α3 ( Aα5 = 3α5 Aα6 = 3α6 + α5    Aα7 = 3α7 Aα8 = 3α8 + α7 Aα9 = 3α9 + α8 Aα10 = 3α10 + α8 There are 4 eigenvectors α1, α3, α5, α7. Notice that 4 is the number of blocks in the Jordan form. 1.2. Properties. Jordan form is the standard form or simple representative in each class given by similar equivalence. AS = SJ, J = diag{J(λ1), · · · J(λk)}, J(λi) = diag{Jl1 (λi), · · · , Jli (λi)} Jj (λi) =        λi 1 λi 1 . . . . . . λi 1 λi        (a) First, J is block diagonal with each diagonal block J(λi) corresponding to one λi . Second, each J(λi) is block diagonal such that the small diagonal blocks all have λi on the diagonal, 1 upper next to diagonal and 0 at other positions. (b) The total dimension of J(λi) is the algebraic multiplicity ni . Corresponding to each λi , the number of small blocks is the geometric multiplicity li . (b) The number and dimensions for those small blocks corresponding to each λi is fully determined by rank(A − λi) s = rs, s = 1, 2, ....... n − r1 = number of blocks corresponding to λi with dimension ≥ 1 × 1 r1 − r2 = number of blocks corresponding to λi with dimension ≥ 2 × 2 r2 − r3 = number of blocks corresponding to λi with dimension ≥ 3 × 3 . . . (n − r1) − (r1 − r2) = number of blocks corresponding to λi with dimension 1 × 1 (r1 − r2) − (r2 − r3) = number of blocks corresponding to λi with dimension 2 × 2 . . . The computation stops at the first si such that rsi = r(si+1). Then corresponding to this λi , the largest block has dimension si × si . (c) Jordan form determines all the properties of A that does not change under similar transformation. For example, eigenvalues, characteristic polynomial, minimal polynomial. (d) Denote S = [α1, ..., αn]. Then those column vectors, which corresponds to the first term of each small block in Jordan form, are eigenvectors