Linear Algebra

Section IV. Jordan Form 393(b) The matrix S is 5×5 with two eigenvalues. For the eigenvalue 2 the nullitiesare: S − 2I has nullity two, and (S − 2I) 2 has nullity four. For the eigenvalue−1 the nullities are: S + 1I has nullity one.2.20 Find the change of basis matrices for each example.(a) Example 2.13 (b) Example 2.14 (c) Example 2.15̌ 2.21 Find ( the Jordan ) form and a Jordan basis for each matrix.−10 4(a)−25 10( )5 −4(b)9 −7( ) 4 0 0(c) 2 1 35 0 4( )5 4 3(d) −1 0 −31 −2 1( )9 7 3(e) −9 −7 −44 4 4( )2 2 −1(f) −1 −1 1−1 −2 2⎛⎞7 1 2 2⎜ 1 4 −1 −1⎟(g) ⎝−2 1 5 −1⎠1 1 2 8̌ 2.22 Find all possible Jordan forms of a transformation with characteristic polynomial(x − 1) 2 (x + 2) 2 .2.23 Find all possible Jordan forms of a transformation with characteristic polynomial(x − 1) 3 (x + 2).̌ 2.24 Find all possible Jordan forms of a transformation with characteristic polynomial(x − 2) 3 (x + 1) and minimal polynomial (x − 2) 2 (x + 1).2.25 Find all possible Jordan forms of a transformation with characteristic polynomial(x − 2) 4 (x + 1) and minimal polynomial (x − 2) 2 (x + 1).̌ 2.26 Diagonalize ( ) these. ( )1 10 1(a)(b)0 01 0̌ 2.27 Find the Jordan matrix representing the differentiation operator on P 3.̌ 2.28 Decide if these two are(similar.) ( )1 −1 −1 04 −3 1 −12.29 Find the Jordan form of this matrix.( )0 −11 0Also give a Jordan basis.2.30 How many similarity classes are there for 3×3 matrices whose only eigenvaluesare −3 and 4?