The given quadratic form is
3x2+5y2+3z2−2xy−2yz+2xz
The matrix of the given quadratic form is
A=⎝⎛3−11−15−11−13⎠⎞
We write , A=IAI
i,e, ⎝⎛3−11−15−11−13⎠⎞ =⎝⎛100010001⎠⎞ A⎝⎛100010001⎠⎞
Now we shall reduce A to diagonal form by applying congruence operation on it . Performing R2→3R2+R1,R3→3R3−R1;C2→C2+31C1,
C3→C3−31C1;R3→7R3+R2;C3→C3−71C2
We get ,
⎝⎛30001400054⎠⎞ = ⎝⎛11−60330021⎠⎞ A⎝⎛100311021−87−11⎠⎞
Performing , R1→31R1,C1→31C1 ;
R2→141R1,C2→141;
R3→541R3,C3→541C2
We get ,
⎝⎛100010001⎠⎞ =⎝⎛ab−6c03b3c0021c⎠⎞A ⎝⎛a003bb021−8c7−cc⎠⎞
Where a=31,b=141,c=541
Thus the linear transformation ,
X=PY where ,
P=⎝⎛a003bb021−8c7−cc⎠⎞
X=[x,y,z]′,Y=[y1,y2,y3]′
This transformation gives quadratic form to the normal form
y12+y22+y32.............(1)
The rank r of the given quadratic form = The number of nonzero terms in its normal form (1)
= 3.
The signature of the given quadratic form = the excess of the number of positive terms over the number of negative terms in its normal form=3−0=3
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