matrix-analysis-w7

實變函數論─應用數學系 吳培元老師  1    Matrix Analysis 第七週課程講義 Nonderogatory matrix: Thm. , Then the following are equiv.: (1) ( ) exactly one Jordan block in Jordan form of ; (2) ( ) for some poly. ; (3) minimal poly. of =ch T n n T T T M p p T                  2 -1 arac. poly. of ; (4) dim Alg ; (5) dim ; (6) dim ker ( - ) 1 ; (7) V , , ,..., ( is cycliC); (8) = ; (9) is abelian; (10) Every matr n n n T T n T n T I x x Tx T x T x T T T T                   11 21 , -1 -1 ix commuting with is a polynomial in T. * (11) , with 0 . 0 j j nn nn T a a T a j a a            ( ) ( ) 1 ( ) 1 ... ( ) ( ) 2 Def: is nonderogatory if it satisfies one of the above. Pf: (3) (1): Jordan form: charac. poly = ( - ) min. poly = ( - ) (1) ... 0 (3) i i ni i i k k i i k i i i n i T x x k k i                2 1 (3) (2): Rational form: charac. poly = (3) ... 1 (2) min. poly = (3) (4): (4) deg. of min. poly.= (3) ( min poly. | charac. poly). (5) (1): j n j p p p p n                ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 2 3 i i i i (5) + 3 +... + 3 +... = =..=0 (1).i i i i i ik k n k k k k i         (1) (6): no. of Jordan blocks at = dim ker ( - )= geom. multi of at ( ) (1) ( ), dimker ( - )=1 ( ), dimker ( - )=0 n n T I T T T T I T T I                    實變函數論─應用數學系 吳培元老師  2      0 1 -1 -1 1 0 -1 2 (2) (7): 0 - 1 - ( ) ( ) if ( ) ... 0 1 - 1 0 Let . 0 0 0 0 1 Then ( ) , ( ) ,.1 0 0 0 0 T n n n n T T a a T M p M p p x x a x a x a a x M p x M p x                                                   n-1 0 .., ( ) 0 1 ( ) is cyclic. But "cyclic" preserved under similarity cyclic. T T M p x M p T             -1 (7) (2): , ,..., basis of .n nx Tx T x   0 0 * 1 0 * represented , , this basis as 0 1 0 0 0 0 1 *       T w i t           0 * 0 1 1 * ( ) for some poly. .0 1 * 0 1 1 * T M p p                                     實變函數論─應用數學系 吳培元老師  3      21 2 2 32 21 1 32 21 1 (11) (7): 1 0 Let . 0 * ** * , ...,0 * 0 0 V , , , . nn n n x a Tx T x T x a a a a a x Tx T x                                                     ( ) ( ) ( ) ( ) ( ) 1 2 1 2 3 -1 1 2 (1) (8): (8) 3 ... = =...=0 (1) (7) (11): , ,..., indep. Gram-Schmidt , ,..., o.n. basis of , where i i i i i i i i n n n k k k i k k i x Tx T x x x x                 1 11 11 2 21 22 22 2 3 31 32 33 33 -1 1 2 , 0 , 0 , 0 ... , 0 n n n n nn nn x b x b x b x b Tx b x b x b Tx b T x b x b x b Tx b T x b                21 1 11 11 2 1 22 22 11 2 2 21 22 1 1 - + bTx b Tx b x x b b b Tx b Tx b T x         31 32 22 3 33 33 33 2 1 1 1 =...+ - - , b bb x x Tx b b b x x x        2 1 , x x  1 2 n 11 22 22 33 , ,..., * * * = 0 0 0 0 T Tx Tx Tx b b b b                    實變函數論─應用數學系 吳培元老師  4                              (8) (9) : Let , Then . (9) (8) : . Let . Then , (8) (8) (10) : ( ) : poly. . = A B T A T AB BA T T A T B T AB BA A T T T T p T p T T                            2 1 Generalizations: (1) , dim Alg( , ) . (M. Garstenhaber, 1961) Note: Alg( , )=V , 0 , -1 dim Alg( , ) . Sketch of pf of (1): A ... with ( ) i j m i A B n n AB BA A B n A B A B i j n A B n A A A            1 1 1 ( ) for . ... (by splitting of commutant). Alg( , ) Alg( , ) ... Alg( , ). 0 1 0 1 May assume 1 ... 1 0 0 j m m m A i j B B B A B A B A B A                                     31 2 1 2 3 1 -1 -1-1 -1 2 2 2 -1 -1 -1 ... Check: V 1, ,..., , , ,..., , , ,..., ,..., , ,..., =Alg l l l k kk k l l l k k k k k k A A B AB A B B AB A B B AB A B       1 ( , ) dim Alg ( , ) +...+ . Ref: J. Barria & P.R. Holmes, Vector bases for two commuting matrices, Linear Multilinear Algebra, 27(1990),147-157. Note1: l A B A B k k n A M     abelian subalegbran  2 1 dim . (I. Schur, 1905) (dim 1 & attained) 4 Note2: ,..., , ,m i j j i A n hA A A n n A A A A i j           1dim Alg ( ,..., ) .mA A n 實變函數論─應用數學系 吳培元老師  5     4 4 1 2 3 4 0 0 0 0 0 0 Ex. V . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c d a I M A A A A                                                         1 4 Then Alg ( ,..., ) & dim 5 4. A A A A        2 2 1 if even, 4 (2) , , rank( - ) 1 dim Alg( , ) 1 ( -1) if odd. 4 Ref: R.M. Guralnick, A note on commuting pairs of matrices, Linear Multilinear Algeb n n n A B n n AB BA A B n n n        ra, 31(1992), 71-75. Eigenvalues of Hermitian matrices. Ref: R.A. Horn & C.R, Johnson, Matrix analysis, Sec. 4.2,4.3. (1) Hermitian Variational characterization of ( ), i.e., max., min. of functions change rate A n n A   1 n = 0. Let ... be eigenvalues of . Note: Eigenvalues of Hermitian are real. Pf: Let . Then , = * , = , || A A Ax x Ax x A x x x Ax       || , , = real x x x x x        實變函數論─應用數學系 吳培元老師  6        1 n 1 1 n Thm1. (Rayleigh-Rite) Hermitian = min , : , 1 = max , : , 1 Pf: Let ,..., be unit eigenvectors corresponding to ... , resp. Note: n n n A n n Ax x x x Ax x x x y y               's may be chosen to be o.n. Reason: (i) Assume , , , || , || , = , j i j i j i i j i i j i j i j j j i j y Ay y y y y y y Ay y y y y               1 1 1 * 1 * * * * 1 * , =0 (ii) If = , then , may be chosen to be o.n. Let U= unitary. Let 1& . , , , n i j i j i j i j n T n n y y y y y y x y U x y y Ax x AUy Uy U AUy y y y U AU A y y y y                                 1 1 1 1 1 * 1 2 ' ' 1 2 2 2 2 1 1 1 1 ' ' ' , , May assume = . min , : 1 = min ... : ... 1 n n n n n n j jj n n n n n y y y y Ax x y y y A Ax x x x x x x                                                                               2 2 2 2 1 1 1 1 max ... : ... 1 Note. Hermitian ( ) , n n n n n x x x x A n n W A x x             實變函數論─應用數學系 吳培元老師  7        1 -1 1 2 Thm2. = min , : 1 & ,... = max , : 1 & ,... 1,..., -1, . Pf:" ": with 1 , , , j j j n n n i i i i j i j n n n n i i k k i i i k k i i j k j i j k j Ax x x x y y Ax x x x y y j n n x a y a Ax x A a y a y a y a y a                               , 2 2 , " " = , RHS Similarly for 2nd equality. Note: Not much use: need eigenvectors to deduce eigenvalues. i k i k i k n n i i j i j i j i j j j j a y y a a Ay y                    -1 -1 1 1 -1 ,...,1 j+1 n,...,1 Thm3. (Courant-Fischer) Hermitian, eigenvalues: ... . Then = max min , : 1, ,..., = min max , : 1, ,..., 1,..., n j n j n j jW W W W A n n Ax x x x W W Ax x x x W W j                   dim - 1 . Another expression: = max min , : 1, || nj M M n j n Ax x x x M            1 -1 dim V ,..., = min max , : 1, || n j M M j W W Ax x x x M         j+1 n V ,...,W W 