(From m.j. shu) known Quadratic Form $ \ Bex f (x, y, z) = x ^ 2 + 3y ^ 2 + Z ^ 2 + 2bxy + 2xz + 2yz \ EEx $ the rank is $2 $. Evaluate the parameter $ B $, and pointed out the equation $ \ Bex f (x, y, z) = 4 \ EEx $ represents what kind of surface?
Answer: matrix by $ F $ \ Bex a =\sex {\ BA {CCC} 1 & B & 1 \ B & 3 & 1 \ 1 & 1 & 1 \ rank of EA} \ RRA \ sex {\ BA {CCC} 1 & B & 1 \ B-1 & 3-B & 0 \ 0 & 1-B & 0 \ EA} \ EEx $$2 $ Zhi $ B = 1 $. in this case, $ \ Bex a =\sex {\ BA {CCC} 1 & 1 & 1 \ 1 & 3 & 1 \ 1 & 1 \ EA }. \ EEx $ by $ \ Bex | \ lm E-A | = \ LM (\ LM-1) (\ lm-4) \ EEx $ Zhi $ F $ can be converted to $ \ Bex F (x_1, x_2, X_3) = Y_2 ^ 2 + 4y_3 ^ 2 through orthogonal linear transformation. \ EEx $ hence $ \ Bex 4 = f (x_1, x_2, X_3) = Y_2 ^ 2 + 4y_3 ^ 2 \ EEx $ is an elliptical cylindrical disk.
[Mathematical Analysis for small readers] (10-14 quadratic form and surface classification)