formes canóniques de diagonalització de àlgebra lineal

A_{ij} = ( < (-a), b >,< (-b),a > )

(-1)·(a+x)·(a+(-x))+b^{2} = ( x^{2}+(-1)·a^{2})+b^{2} = 0

x = ( a^{2}+(-1)·b^{2} )^{(1/2)}

Forma de canónica:

( a^{2}+(-1)·b^{2} )^{(1/2)}·Id_{22} = ...

... ( < ( a^{2}+(-1)·b^{2} )^{(1/2)},0 >,< 0,( a^{2}+(-1)·b^{2} )^{(1/2)} > )

(-1)·( a+( a^{2}+(-1)·b^{2} )^{(1/2)} )·( a+(-1)·( a^{2}+(-1)·b^{2} )^{(1/2)} )+b^{2} = 0

Base canónica:

X_{ij} = ...

... ( < a+(-1)·( a^{2}+(-1)·b^{2} )^{(1/2)} , (-b) >,< (-b) , a+( a^{2}+(-1)·b^{2} )^{(1/2)} > )

det[A] = 2b^{2}

Inversa de la base canónica:

( X_{ij} )^{o(-1)} = ( 1/(2b^{2}) )·...

... ( < a+( a^{2}+(-1)·b^{2} )^{(1/2)} , b >,< b , a+(-1)·( a^{2}+(-1)·b^{2} )^{(1/2)} > )

Cálculo provisional:

( X_{ij} )^{o(-1)} o A_{ij} = ...

( < (-a)·( a+( a^{2}+(-1)·b^{2} )^{(1/2)} )+(-1)·b^{2},...

... b·( a^{2}+(-1)·b^{2} )^{(1/2)} >,

< b·( a^{2}+(-1)·b^{2} )^{(1/2)} , ...

... b^{2}+a·( a+(-1)·( a^{2}+(-1)·b^{2} )^{(1/2)} ) > )

( X_{ij} )^{o(-1)} o A_{ij} o X_{ij} = ( a^{2}+(-1)·b^{2} )^{(1/2)}·Id_{22}

A_{ij} = ( < (-a),b >,< (-b),a > )

Composición Nil-potente:

( < (-a)+(-x) , b >,< (-b), a+(-y) > ) o ( < (-a)+(-x) , b >,< (-b), a+(-y) > ) = ...

... ( < ( (-a)+(-x) )^{2}+(-1)·b^{2} , b·((-a)+(-x))+b·(a+(-y) ) >,...

... < (-b)·((-a)+(-x))+(-b)·(a+(-y)), ( a+(-y) )^{2}+(-1)·b^{2} > ) = 0_{ij}

x = (-a)+b & y = a+(-b)

Base nil-potente:

( < (-b),b >,< (-b),b > )^{o1} ==> <x,x>

( < (-b),b >,< (-b),b > )^{o2} ==> <x,y> & x != y

Y_{ij} = ( < x,x>,<x,y> )

Z_{ij} = ( < y,x >,< x,x > )

det[Y_{ij}] = det[Z_{ij}]

A_{ij} o Y_{ij} = ...

... ( < (-a)·x+b·x , (-a)·x+by >,< (-b)·x+ax , (-b)·x+ay > )

Z_{ij} o A_{ij} o Y_{ij} = ...

... ( < ( (-a)+b )·det[Y_{ij}] , b·(y^{2}+(-1)·x^{2}) >,< 0,(a+b)·det[Y_{ij}] > )

Base de Jordan:

y = ( x^{2}+(1/b) )^{(1/2)}

( < x,x >,< x,( x^{2}+(1/b) )^{(1/2)} > )

Z_{ij} o A_{ij} o Y_{ij} = ...

... ( < (-a)·det[Y_{ij}],0 >,< 0,a·det[Y_{ij}] > )+( < b·det[Y_{ij}],1 >,< 0,b·det[Y_{ij}] > )

A_{ij} = ( <a,b>,<b,a> )

Composición Nil-potente:

( < a+(-x) , b >,< b, a+(-y) > ) o ( < a+(-x) , b >,< b, a+(-y) > ) = ...

... ( < ( a+(-x) )^{2}+b^{2} , b·(a+(-x))+b·(a+(-y) ) >,...

... < b·(a+(-x))+b·(a+(-y)), ( a+(-y) )^{2}+b^{2} > ) = 0_{ij}

x = a+ib & y = a+(-1)·ib

Base nil-potente:

( < (-1)·ib , b >,< b, ib > )^{o1} ==> < x , ix >

( < (-1)·ib , b >,< b, ib > )^{o2} ==> < ix , y > & y != (-x)

Y_{ij} = ( < x,ix>,<ix,y> )

Z_{ij} = ( < y,(-i)x >,< (-i)x,x > )

det[Y_{ij}] = det[Z_{ij}]

A_{ij} o Y_{ij} = ...

... ( < ax+bix , aix+by >,< bx+aix , bix+ay > )

Z_{ij} o A_{ij} o Y_{ij} = ...

... ( < ( a+bi )·det[Y_{ij}] , b·(y^{2}+x^{2}) >,< 0,(a+bi)·det[Y_{ij}] > )

Base de Jordan:

y = ( (-1)·x^{2}+(1/b) )^{(1/2)}

( < x,ix>,<ix,( (-1)·x^{2}+(1/b) )^{(1/2)}> )

( (-1)·x^{2}+(1/b) )^{(1/2)} = (-x)

x != (1/2b)^{(1/2)} & x != (-1)·(1/2b)^{(1/2)}

Z_{ij} o A_{ij} o Y_{ij} = ...

... ( < a·det[Y_{ij}],0 >,< 0,a·det[Y_{ij}] > )+( < bi·det[Y_{ij}],1 >,< 0,bi·det[Y_{ij}] > )

A_{ij} = ( <a,b>,<a,a> )

Composición Nil-potente:

( < a+(-x) , b >,< a, a+(-y) > ) o ( < a+(-x) , b >,< a, a+(-y) > ) = ...

... ( < ( a+(-x) )^{2}+ba , b·(a+(-x))+b·(a+(-y) ) >,...

... < a·(a+(-x))+a·(a+(-y)), ( a+(-y) )^{2}+ab > ) = 0_{ij}

x = a+i·(ab)^{(1/2)} & y = a+(-i)·(ab)^{(1/2)}

Base nil-potente:

( < (-1)·i·(ab)^{(1/2)} , b >,< a , i·(ab)^{(1/2)} > )^{o1} ==> < b·x , (ab)^{1/2})·ix >

( < (-1)·i·(ab)^{(1/2)} , b >,< a , i·(ab)^{(1/2)} > )^{o2} ==> ...

... < (ab)^{(1/2)}·x , y > & y != a·ix

Y_{ij} = ( < bx,(ab)^{(1/2)}·ix >,< (ab)^{(1/2)}·x,y > )

Z_{ij} = ( < y,(-1)·(ab)^{(1/2)}·ix >,< (-1)·(ab)^{(1/2)}·x,bx > )

det[Y_{ij}] = det[Z_{ij}]

A_{ij} o Y_{ij} = ...

... ( < a·bx+b·(ab)^{(1/2)}·x, a·(ab)^{(1/2)}·ix+by >,...

... < a·bx+a·(ab)^{(1/2)}·x, a·(ab)^{(1/2)}·ix+ay > )

Z_{ij} o A_{ij} o Y_{ij} = ...

... ( < ( a+(ab)^{(1/2)} )·det[Y_{ij}], b(y^{2}+(-1)·a^{2}·x^{2}) >,...

... < 0, ( a+(-1)·(ab)^{(1/2)} )·det[Y_{ij}] > )

Base de Jordan:

y = ( a^{2}x^{2}+(1/b) )^{(1/2)}

( < bx,(ab)^{(1/2)}·ix >,< (ab)^{(1/2)}·x,( a^{2}x^{2}+(1/b) )^{(1/2)} > )

a^{2}x^{2}+(1/b) = (-1)·a^{2}·x^{2}

x != (i/a)·(1/2b)^{(1/2)} & x != (-1)·(i/a)·(1/2b)^{(1/2)}

Z_{ij} o A_{ij} o Y_{ij} = ...

... ( < (ab)^{(1/2)}·det[Y_{ij}], 0 >,< 0,(-1)·(ab)^{(1/2)}·det[Y_{ij}] > )+...

... ( < a·det[Y_{ij}], 1 >,< 0,a·det[Y_{ij}] > )

A_{ij} = ( <a,0,b>,<0,a,0>,<b,0,a> )

Composición Nil-potente:

( < a+(-x),0,b >,< 0,a+(-y),0 >,< b,0,a+(-z) > ) o ...

... ( < a+(-x),0,b >,< 0,a+(-y),0 >,< b,0,a+(-z) > o ...

... ( < a+(-x),0,b >,< 0,a+(-y),0 >,< b,0,a+(-z) > ) = ...

... ( < ( a+(-x) )^{2}+b^{2} , 0 , b·(a+(-x))+b·(a+(-z) ) >,...

... < 0 , (a+(-y))^{2} , 0 >,...

... < b·(a+(-x))+b·(a+(-z)) , 0 , ( a+(-z) )^{2}+b^{2} > ) o ...

... ( < a+(-x),0,b >,< 0,a+(-y),0 >,< b,0,a+(-z) > ) = 0_{ij}

x = a+ib & y = a & z = a+(-i)·b

Base nil-potente:

( < ix,0,x >,< 0,y.0 >,< ix,0,z > ) & ( y != x || z != x )

Y_{ij} = ( < ix,0,x >,< 0,y.0 >,< ix,0,z > )

Z_{ij} = ( < z,0,(-x) >,< 0,y.0 >,< (-i)·x,0,ix > )

det[Y_{ij}] = det[Z_{ij}]

A_{ij} o Y_{ij} = ...

... ( < a·ix+bix,0,ax+bz >,< 0,ay.0 >,< b·ix+aix,0,bx+az > )

Z_{ij} o A_{ij} o Y_{ij} = ...

... ( < (a+b)·(1/y)·det[Y_{ij}],0,b·(z^{2}+(-1)·x^{2}) >,...

... < 0,ay^{2},0 >,...

... < 0,0,(a+(-b))·(1/y)·det[Y_{ij}] > )

Base de Jordan:

y = 1

z = ( x^{2}+(1/b) )^{(1/2)}

( < ix,0,x >,< 0,1,0 >,< ix,0,( x^{2}+(1/b) )^{(1/2)} > )

Z_{ij} o A_{ij} o Y_{ij} = ...

... ( < b·det[Y_{ij}],0,0 >,< 0,a,0 >,< 0,0,(-b)·det[Y_{ij}] > )+ ...

... ( < a·det[Y_{ij}],0,1 >,< 0,0,0 >,< 0,0,a·det[Y_{ij}] > )