ВЕКТОРНОЕ ПРЕДСТАВЛЕНИЕ АССОЦИАТИВНЫХ ПРОИЗВЕДЕНИЙ СОПРЯЖЕННЫХ КВАТЕРНИОННЫХ МАТРИЦ

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  • 4

    - 1/4 ( 49 ) 2011

    -

    512.643.8:531.3

    . . ,

    . . , 19, . , , 49027 .: 067-72-607-72

    . .

    * .: 050-321-14-60

    . .

    * .: 067-921-10-67; (056) 713-58-03*

    .

    . , 2, . , , 49010

    - - . , - - -

    : , , ,

    - - . , - - -

    : -, , - , -

    Using mathematical induction the procedu-re of vector representation of adjoint quatern-ionic matrices associative products were built. Some symbolic formulas that shows an equiv-alence of associative products of quaternion-ic matrices and multiplicative compositions of vector algebra

    Key words: quaternionic matrices, multipli-cative compositions, associative products, vec-tor presentation

    [3], -, , [1, 8], , - -[2]. -- [5, 6]. - . -

    , - , - .

    a b, , a b c, , , a b c d, , , , A At0 0, ( a0 0= ). -

  • 5

    -

    , , , - .

    a b a b a b a b1 1 2 2 3 3+ + = , .. a a ab

    b

    b

    a b1 2 3

    1

    2

    3

    =

    a b a b a b2 3 3 2 1 = ( )a b a b a b3 1 1 3 2 = ( )a b a b a b1 2 2 1 3 = ( ) ,

    ..[7]

    0

    0

    0

    3 2

    3 1

    2 1

    1

    2

    3

    1

    2

    3

    =

    ( )( )( )

    = a a

    a a

    a a

    b

    b

    b

    a b

    a b

    a b

    a b

    :

    A b

    a b

    a b

    a b

    a b

    0 0

    1

    2

    3

    ( )( )( )

    , A b

    a b

    a b

    a b

    a b

    t0 0

    1

    2

    3

    ( ) ( ) ( )

    ,

    [5]

    A ba b

    a b0 0

    , A b

    a b

    a bt0 0

    ( ) ,

    A At0 0, - a0 0= :

    A

    a a a

    a a a

    a a a

    a a a

    0

    1 2 3

    1 3 2

    2 3 1

    3 2 1

    0

    0

    0

    0

    =

    , A

    a a a

    a a a

    a a a

    a a a

    t0

    1 2 3

    1 3 2

    2 3 1

    3 2 1

    0

    0

    0

    0

    =

    .

    - a b c, , - [5]:

    A B c A B c0 0 0 0 0 0 ( ) = ( ) ,

    A B c A B ct t0 0 0 0 0 0 ( ) = ( ) ,A B c A B ct t0 0 0 0 0 0 ( ) = ( ) ,A B c A B ct t t t0 0 0 0 0 0 ( ) = ( ) ,

    :

    :

    A B ca b c

    a b c a b c0 0 0 ( )

    ( ) ( ) + ( ) .

    [5]:

    A B ca b c

    a b c a b c

    A B ca b c

    a b

    t

    t

    0 0 0

    0 0 0

    ( ) ( ) ( ) ( )

    ( ) ( )

    ,

    ( ) ( ) ( ) ( )

    ( ) + ( )

    c a b c

    A B ca b c

    a b c a b ct t

    ,

    .0 0 0

    A B c0 0 0( ) [4], :

    A B

    a b a b a b a b

    a b a b a b a b

    a0 0

    1 2 3

    1 3 2 =

    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

    b a b a b a b

    a b a b a b a b

    2 3 1

    3 2 1

    .

    A B c

    a b c a b c a b c

    a b c a b c a b0 0 0

    1 1 2 2 3 3

    1 3 2( ) =

    ( ) + ( ) + ( ) ( ) ( ) + (( ) ( ) + ( ) ( ) ( ) ( ) + ( )

    2 3

    2 3 1 1 3

    3 2 1 1 2

    c

    a b c a b c a b c

    a b c a b c a b c

    A B ca b c

    a b c a b c0 0 0( )

    ( ) ( ) + ( ) .

    A B ct0 0 0( ) [4],

    A B c

    a a a

    a a a

    a a a

    a a a

    b c

    b c

    b c0 0 0

    1 2 3

    1 3 2

    2 3 1

    3 2 1

    1

    0

    0

    0

    0

    ( ) =

    ( )( ))( )

    =

    ( ) + ( ) + ( ) ( ) ( ) +

    2

    3

    1 1 2 2 3 3

    1 3 2 2

    b c

    a b c a b c a b c

    a b c a b c a b cc

    a b c a b c a b c

    a b c a b c a b c

    ( ) ( ) + ( ) ( ) ( ) ( ) + ( )

    3

    2 3 1 1 3

    3 2 1 1 2

    A B

    a b a b a b a b

    a b a b a b a bt

    0 0

    1 2 3

    1 1 12

    =

    ( ) ( ) ( ) ( ) ( ) ( ) ( )

    33 2 1 2 3 1

    2 3 1 2 2 2 1

    2 2

    2 2

    ( ) ( ) ( ) ( ) ( )

    a b a b a b

    a b a b a b a b a b a b 22

    2 2 2

    3 2

    3 2 1 3 1 2 3 3 3

    a b

    a b a b a b a b a b a b a b ( ) ( ) ( ) ( )

  • 6

    - 1/4 ( 49 ) 2011

    :

    , , :

    A B c

    a b c

    a b c a c a c a c b a b ct

    0 0 0

    1 1 1 2 2 3 3 12( ) =

    ( )( ) + +( ) + ( ) ( ) + +( ) + ( ) ( ) +

    1

    2 1 1 2 2 3 3 22

    3 1 1

    2

    2

    a b c a c a c a c b a b c

    a b c a c a22 2 3 3 33

    c a c b a b c+( ) + ( ) .. :

    A B ca b c

    a b c a c b a b ct

    0 0 02

    ( ) ( )( ) ( ) + ( ) .

    :

    A B ca b c

    a b c a c b a b c

    A B ca

    t

    t t

    0 0 0

    0 0 0

    2( ) ( )

    ( ) ( ) + ( )

    ( )

    ,

    bb c

    c a b a b c

    ( ) ( ) + ( ) .

    - a b c d, , , - - [5, 6]:

    1. A B C d A B C d A B C d

    A

    0 0 0 0 0 0 0 0 0 0 0 0

    0

    ( ) = ( ) = ( ) == BB C d A B C d0 0 0 0 0 0 0( ) = ( ) ( ),

    2. A B C d A B C d A B C dt t t0 0 0 0 0 0 0 0 0 0 0 0 ( ) = ( ) = ( ) == AA B C d A B C dt t0 0 0 0 0 0 0 0 ( ) = ( ) ( ),

    3. A B C d A B C d A B C dt t t0 0 0 0 0 0 0 0 0 0 0 0 ( ) = ( ) = ( ) == AA B C d A B C dt t0 0 0 0 0 0 0 0 ( ) = ( ) ( ),

    4. A B C d A B C d A B C dt t t t t t0 0 0 0 0 0 0 0 0 0 0 ( ) = ( ) = ( ) 00

    0 0 0 0 0 0 0 0

    =

    = ( ) = ( ) ( )A B C d A B C dt t t t ,

    5. A B C d A B C d A B C dt t t0 0 0 0 0 0 0 0 0 0 0 0 ( ) = ( ) = ( ) == AA B C d A B C dt t0 0 0 0 0 0 0 0 ( ) = ( ) ( ),

    6. A B C d A B C d A B C dt t t t t t0 0 0 0 0 0 0 0 0 0 0 ( ) = ( ) = ( ) 00

    0 0 0 0 0 0 0 0

    =

    = ( ) = ( ) ( )A B C d A B C dt t t t ,

    7.

    A B C d A B C d

    A B C

    t t t t

    t t

    0 0 0 0 0 0 0 0

    0 0 0

    ( ) = ( ) == ( ) dd A B C d

    A B C d

    t t

    t t

    0 0 0 0 0

    0 0 0 0

    = ( ) == ( ) ( ),

    8.

    A B C d A B C d

    A B C

    t t t t t t

    t t t

    0 0 0 0 0 0 0 0

    0 0 0

    ( ) = ( ) == ( ) = ( ) == ( ) ( )

    d A B C d

    A B C d

    t t t

    t t t

    0 0 0 0 0

    0 0 0 0 ,

    - - , - . A B C d0 0 0 0 ( ) , - :

    A B C d

    a a a

    a a a

    a a a

    a a a

    b c d

    0 0 0 0

    1 2 3

    1 3 2

    2 3 1

    3 2 1

    10

    0

    0

    0

    ( ) =

    =

    (( ) + ( ) + ( ) ( ) ( ) + ( ) ( ) +

    1 2 2 3 3

    1 3 2 2 3

    2

    b c d b c d

    b c d b c d b c d

    b c d b33 1 1 3

    3 2 1 1 2

    c d b c d

    b c d b c d b c d

    ( ) ( ) ( ) ( ) + ( )

    :

    A B C d

    a b c d a b c d a b c d

    a

    0 0 0 0

    1 1 11

    2 2

    2

    ( ) =

    =

    ( ) + ( ) ( ) ++

    bb c d a b c d a b c d

    a b c d a b c d

    ( ) ( ) + ( )

    ( ) (

    23 3 3

    3

    1 1 1 1 2)) ( ) + ( )

    ( ) ( ) + 2 1 3 3 3 2

    32

    2 3 2

    a b c d a b c d

    a b c d a b c d a b c d(( )

    ( ) ( ) ( ) ( ) ++

    3

    2 1 1 2 2 2 2 3 3 3 1

    3

    a b c d a b c d a b c d a b c d

    a b c ( ) + ( ) ( )

    ( ) ( )

    d a b c d a b c d

    a b c d a b c d

    11 3 1

    3

    3 1 1 3 2 2aa b c d a b c d

    a b c d a b c d a b c d

    3 3 3 2 1

    21

    1 2 1

    ( ) + ( ) ( ) ( ) + ( ) 2

    , , - :

    A B C d

    a b c d a b c d

    a b c d a

    0 0 0 0( )

    ( ) ( ) + ( )

    ( ) bb c d a b c d( ) ( ) + ( ) ;

    :

    A B c

    a b c a b c a b c

    a b c a b c a bt

    0 0 0

    1 1 2 2 3 3

    1 1 1 12( ) =

    ( ) ( ) ( )( ) (( ) + ( ) ( ) + ( )

    3 2 2 1 2 2 3 3 1 3

    2 2 2 2 3 1

    2 2

    2 2

    c a b c a b c a b c

    a b c a b c a b c a11 2 1 1 3 3 2 3

    3 3 3 3 2 1 1 3 1

    2

    2 2

    b c a b c a b c

    a b c a b c a b c a b c a

    ( ) ( ) ( ) + bb c a b c( )

    1 2 2 3 22

  • 7

    -

    A B C d

    b c a d a b c d

    a b c d a

    0 0 0 0( )

    ( ) ( ) + ( )

    ( ) dd b c a b c d( ) ( ) + ( ) ;

    A B C d

    a b c d a b c d

    a b c d a

    0 0 0 0( )

    ( ) ( ) + ( )

    ( ) bb c d a b c d( ) ( )+ ( ) ;

    A B C d

    b c a d a b c d

    a b c d b

    0 0 0 0( )

    ( ) ( ) + ( )

    ( ) cc a d a b c d( ) ( )+ ( ) ;

    A B C d

    a b c d a b c d

    a b c d a

    t0 0 0 0( )

    ( ) ( ) ( )

    ( ) bb c d a b c d( ) ( ) ( ) ;

    A B C d

    a b c d a b c d

    a b c d a

    t0 0 0 0( )

    ( ) ( ) ( )

    ( ) ( ) ( ) ( ) b c d a b c d;

    A B C d

    a b c d a b c d

    a b c d a

    t t0 0 0 0( )

    ( ) ( ) + ( )

    ( ) ( ) ( ) + ( ) b c d a b c d;

    A B C d

    b c a d a b c d

    a b c d a

    t t0 0 0 0( )

    ( ) ( ) + ( )

    ( ) ( ) ( ) + ( ) d b c a b c d;

    A B C d

    a b c d a b c d

    a b c d a

    t0 0 0 0( )

    ( ) ( ) + ( )

    ( ) + ( ) ( ) ( ) b c d a b c d;

    A B C d

    b c a d a b c d

    a b c d a

    t0 0 0 0( )

    ( ) ( ) + ( )

    ( ) + ( ) ( ) ( ) d b c a b c d;

    A B C d

    a b c d a b c d

    a b c d a

    t t0 0 0 0( )

    ( ) ( ) ( )

    ( ) + ( ) ( ) + ( ) b c d a b c d;

    A B C d

    a b c d a b c d

    a b c d

    t t0 0 0 0( )

    ( ) ( ) ( )

    ( ) + aa b c d a b c d( ) ( ) + ( ) ;

    A B C d

    a b c d a b c d

    a b c d

    t t t0 0 0 0( )

    ( ) ( ) + ( )

    ( )