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二次形式主轴1注意形式2要会进行矩阵转换相同请注意以下之推导
SEC. 8.4 Eigenbases. DiagonaUzation. lll~.If~,i-I Forms
By definition, a £llllllOlnll~Oilll£ form Q in the components Xl,· .. ,X n of a vector x is a sum
2
of n terms, namely,
A == [ajk ] is called the coefficient matrix of the form. We may assume that A is
because we can take off-diagonal terms together in and write the result
as a sum of two terms; see the tr\.IIr\.,(iT1I1I4,,n- v..I-.U.AJlAl-/AV.
Let
Here 4 + 6 = 10 = 5 + 5. From the corresponding symmetric matrix C = [Cjk] , where Cjk = iCajk + akj) ,
thus Cll = 3, Cl2 = C21 = 5, C22 = 2, we get the same result; indeed,
Uu-aOJratJlC forms occur in physics and ;;,vJJlJlJlvIl.-Jl Instan.ce, in connection with conic
2
sections xII a + x~/b2 == 1, Their
transformation to 11\1111-111.01111\111 axes is an 1Irtf\1I\r\.1Itl1lnt 1I\-rl1.0tl.0 :lI
of matrices, as follows.
Theorem 2, the coefficient matrix A of has an orthonormal basis of
eigenvectors. Hence if we take these as column vectors, we obtain a
orthogonal, so that .From we thus have A ==
into (7)
(8
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