二次型 f = 3x^2 +5y^2+5z^2+4xy-4xz-10yz =1的矩陣 A =
[ 3 2 -2]
[ 2 5 -5]
[-2 -5 5]
|λE-A| =
|λ-3 -2 2|
|-2 λ-5 5|
| 2 5 λ-5|
| 0 λ λ|
|λ-3 -4 2|
|-2 λ-10 5|
| 0 0 λ|
= λ(λ-2)(λ-11), 得特徵值 λ = 0, 2， 11.
對應的特徵向量依次為(0, 1, 1)^T, (4, -1, 1)^T, (1, 2, -2)^T,
單位化是[0, 1/√2, 1/√2]^T, [4/(3√2), -1/(3√2), 1/(3√2)]^T, (1/3, 2/3, -2/3)^T,
以它們為列組成正交矩陣 Q， 記 向量 p = (x, y, z)^T, q = (u, v, w)^T, 且 p = Qq，
因 f = (p^T)Ap,
則 f = (qQ)^T A (Qq) = q^T(Q^TAQ)q = q^T ∧ q
= q^T diag(0, 2, 11) q = 2v^2 + 11w^2 = 1, 二次型的標準形式是柱面。
二次型 f = 3x^2 +5y^2+5z^2+4xy-4xz-10yz =1的矩陣 A =
[ 3 2 -2]
[ 2 5 -5]
[-2 -5 5]
|λE-A| =
|λ-3 -2 2|
|-2 λ-5 5|
| 2 5 λ-5|
|λE-A| =
|λ-3 -2 2|
|-2 λ-5 5|
| 0 λ λ|
|λE-A| =
|λ-3 -4 2|
|-2 λ-10 5|
| 0 0 λ|
= λ(λ-2)(λ-11), 得特徵值 λ = 0, 2， 11.
對應的特徵向量依次為(0, 1, 1)^T, (4, -1, 1)^T, (1, 2, -2)^T,
單位化是[0, 1/√2, 1/√2]^T, [4/(3√2), -1/(3√2), 1/(3√2)]^T, (1/3, 2/3, -2/3)^T,
以它們為列組成正交矩陣 Q， 記 向量 p = (x, y, z)^T, q = (u, v, w)^T, 且 p = Qq，
因 f = (p^T)Ap,
則 f = (qQ)^T A (Qq) = q^T(Q^TAQ)q = q^T ∧ q
= q^T diag(0, 2, 11) q = 2v^2 + 11w^2 = 1, 二次型的標準形式是柱面。