# Hence c f is similar to diag c p 1 c p s it then

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Hence, C ( f ) is similar to diag( C ( p 1 ) , · · · , C ( p s )). It then follows, by the similarity of direct sums and shuffling, that diag( C ( f k ) , · · · , C ( f n )) is similar to diag( C ( d 1 ) , · · · , C ( d m )). 3. Since both the similarity invariants of C ( q e i i ) and that of C e i ( q i ) are equal to { 1 , · · · , 1 , q e i i } , C ( q e i i ) is similar to C e i ( q i ). Hence diag C ( d 1 ) , · · · , C ( d m ) is similar to diag C e 1 ( q 1 ) , · · · , C e m ( q m ) . Example 5.5. Suppose the elementary divisors of A M 5 ( Q ) are x 2 - 2 , x - 1 , ( x - 1) 2 . Then its similarity invariants are ( x - 1) and ( x 2 - 2)( x - 1) 2 = x 4 - 2 x 3 - x 2 + 4 x - 2 . Then matrix xI - A is equivalent over F [ x ] , for any field F containing Q , to diag( x - 1 , 1 , 1 , 1 , x 4 - 2 x 3 - x 2 + 4 x - 2) and diag( x - 1 , 1 , ( x - 1) 2 , 1 , x 2 - 2) . It is equivalent over R [ x ] to diag( x - 2 , x + 2 , x - 1 , 1 , ( x - 1) 2 ) .
5.7. SIMILARITY 171 Hence, over any field containing Q ,, A is similar to the following matrices 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 2 - 4 1 2 , 1 0 0 0 0 0 0 1 0 0 0 - 1 2 0 0 0 0 0 0 1 0 0 0 2 0 Over R , it is similar to 2 0 0 0 0 0 - 2 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 - 1 2 Example 5.6. Suppose q = x - a is linear. Then C ( q ) = ( a ) so that C e ( q ) = a 1 0 · · · 0 0 0 a 1 · · · 0 0 . . . . . . . . . . . . . . . 0 0 0 · · · a 1 0 0 0 · · · 0 a =: J e ( a ) is indeed a Jordan block. In particular when F = C , the canonical form of the direct sum of hyper-companion matrices of elementary divisors stated in the Theorem is in fact the Jordan form. Example 5.7. Suppose F = R and q = x 2 - bx - c with b 2 + 4 c < 0 so that q irreducible. Then C ( q ) = 0 1 c b . For e = 3 , C 3 ( q ) = 0 1 0 0 0 0 c b 1 0 0 0 0 0 0 1 0 0 0 0 c b 1 0 0 0 0 0 0 1 0 0 0 0 c b , C ( q 3 ) = 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 c 3 3 bc 2 3 c ( b 2 - 1) b 3 - 6 bc 3( c - b 2 ) 3 b Example 5.8. Suppose F = R and q ( x ) = ( x - λ )( x - ¯ λ ) where λ = α + i β , β > 0 , α R . Consider C = C ( α, β ) := α β - β α 2 × 2 , C k = C k ( α, β ) := C I 0 · · · 0 0 0 C I · · · 0 0 . . . . . . . . . . . . . . . 0 0 0 · · · C I 0 0 0 · · · 0 C (2 k ) × (2 k ) . Then det( xI - C ) = q , and since β > 0 , xI - C is equivalent over R [ x ] to diag(1 , q ) . Also, det( xI - C k ) = (det( xI 2 - C )) k = q k . Hence, the minimum polynomial of C k is q l for some integer l k . One can calculate q ( C ) = 0 I 2 and q ( C k ) = 0 2 βE I 0 . . . . . . . . . 0 2 βE I 0 2 βE 0 0 , E = 0 1 - 1 0 .
172 CHAPTER 5. THEORY OF MATRICES It is then easy to calculate that q ( C k ) k - 1 = 0 (2 βE ) k - 1 0 I 2( k - 1) 0 6 = 0 I. Thus, the minimum polynomial of C k is q k . Hence, the companion matrix C ( q k ) is similar to the real Jordan matrix C k ( α, β ) . From this, we obtain the real Jordan canonical form. Corollary 5.3. Every square complex matrix is similar to a Jordan form diag J e 1 ( λ 1 ) , · · · , J e m ( λ m ) . Every square real matrix is similar over R to a real Jordan form diag J e 1 ( λ 1 ) , · · · , J e k ( λ k ) , C e k +1 ( α k +1 , β β k +1 ) , · · · , C e m ( α m , β m ) where λ i , i = 1 , · · · , k are real eigenvalues, and α j + β j i are complex eigenvalues.
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