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F R C Fn

, Mnm(F) F n

m

Fn v =

|v

|

.

m n

A Mnm(F) A = U P U Mnm(F) P Mmm(F)

F= R n= m

T : Rn Rn

AA A U P

A

P=

AA

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AA AA Mmm(F)

vFm

0 Av2 =Av,Av=v, AAv= vAAv .

A

A

A

A F AA = W DW W D

AA

D=

21

2m

,

0 1, . . . , mR A

AA= W

1

m

W .

AA

2= AA

AA

AA

vFm

Av2 =Av,Av=v, AAv= v, AA AAv= AAv, AAv= AAv2 ,

Av=

AAv . A

AA

ker A= ker

AA .

m= dim ker A + A= dim ker

AA +

AA

AA= A .

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AA

1, . . . , m F

m

AA

AA

AA i = ii i= 1, . . . , m

1 . . . > 0 m = A

+1, . . . , m

ker A = ker A

A

A

A= U P P = AA U Mnm(F)

1

1A1, . . . ,

1

A 1 i, j

1

iAi ,

1

jAj

=

1

ijAi, Aj= 1

iji, AAj= 1

ij

i,

2jj

=

j

ii, j= i,j

1, . . . m

+1, . . . , m ker A

1

1A1, . . . ,

1

A A

m n < m +1, . . . , m

1

1A1, . . . ,

1

A, +1, . . . , m F

n

m n.

U=

| | | |1

1A1 1 A +1 m| | | |

1 2

m

Mnm(F) .

U

U

1 2

m

|i

|

=

0

0

1

0

0

i ,

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U i=

1

iAi 1 i

i < i m

U P i =

iU i 1 i

0 U i < i m=

Ai 1 i

0 i < i m=Ai

Ai = 0 i = + 1, . . . , m

ker A A= U P 1, . . . , m

A ker A= ker

AA

AA U

U=A

P1

A Mnm(F) m n A Mmn(F) U Mmn(F) A = U AA A= AA U AA

U

n = m

A

U, U Mnn(F)

A= U

AA=

AA U .

U=U

A AA =AA

A Mnm(F)

A= V D W ,

V Mnp(F) W Mmp(F) p= min {m, n}

D Mpp(F)

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A Mnm(F) A= V D W V Mnn(F)

W Mmm(F) D Mnm(F)

D Mnm(F) (D11, . . . , Dnn)

V =

| |v1 vp| |

; W =

| |w1 wp| |

; D=

d1 0

0 dp

.

W

W =V

V = Ip

1 i p

AA wi= W DVV DW wi= W D

2W wi= W D2

w1

wp

|wi

|

=d2i wi

AA vi= V DWW DV vi= V D

2V vi= V D2

v1

vp

|vi

|

=d2i vi .

wi W AA

d2i m n p = m

W = | |1 m

| |

; D= 1 0

0 m

.

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m n p = m A Mnm(F)

A = U AA= U W

1

m

W =

=

| | | |1

1A1 1 A +1 m| | | |

W W

1

m

W =

=

| | | |

1

1A1 1 A +1 m| | | |

V

1

m

D

1

m

W

;

D =

1

m

V =

| | | |1

1A1 1 A +1 m| | | |

W =

| |1 m| |

A

vi V

AA 2i 1 i

AAvi= AA

1

iAi=

1

iA (AAi) =

1

iA

2i i

= 2i

1

iAi

.

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+ 1 i m +1, . . . , m( A) ( A) ker AA v( A) uFn

0 =AAu, v=u,AAv

AAv= 0 + 1 i m

i= 0 v

i=

iker AA

AAvi = 0 = 0 i= 2i i .

vi V AA

2i