正規直交化法 問題1 解答 - 熊本大学

正規直交化法 問題 1 ^解答

・ 正規直交化法 用 , 次 基底 正規直交基底 作 .

(1) a1 =



 1 0 0



, a2 =



 1 2

−1



, a3 =





−1 1 2





[解]: 与 基底 直交基底 .

b₁ =a₁ =



 1 0 0



,

b₂ =a₂− (a₂, b₁) (b1, b1)b₁ =



 1 2

−1



−



 1 0 0



=



 0 2

−1



,

b3 =a3− (a₃, b₁)

(b₁, b₁)b1− (a₃, b₂) (b₂, b₂)b2 =





−1 1 2



−





−1 0 0



−



 0 0 0



=



 0 1 2



.

直交基底 b1, b2, b3 正規化 行 正規直交基底 求 . ,求 正規直交基底c₁, c₂, c₃ ,

c₁ = 1



 1 0 0



=



 1 0 0



, c₂ =

√5 5



 0 2

−1



=



 0

2√ 5

−5^√₅⁵



, c₃ =

√5 5



 0 1 2



=





√0

5 5 2√

5 5



.

(2) a₁ =



 0 1 1



, a₂ =



 1 0 1



, a₃ =



 1 1 0





[解]: 与 基底 直交基底 .

b₁ =a₁ =



 0 1 1



,

b2 =a2−(a₂, b₁) (b₁, b₁)b1 =



 1 0 1



−



 0

1 2 1 2



=



 1

−¹₂

1 2



,

b₃ =a₃−(a₃, b₁)

(b₁, b₁)b₁− (a₃, b₂) (b₂, b₂)b₂ =



 1 1 0



−



 0

1 2 1 2



−





1

−3¹₆

1 6



=





2 3 2

−3²₃



.

直交基底 b₁, b₂, b₃ 正規化 行 正規直交基底 求 . ,求 正規直交基底c1, c2, c3 ,

c1 =

√2 2



 0 1 1



=





√0

2

√2 2 2



, c2 =

√6 3



 1

−¹₂

1 2



=





√6 3

−_√^√₆⁶

6 6



, c3 =

√3 2





2 3 2

−3²₃



=





√3

√3 3

−3^√₃³



.

(3) a₁ =



 1 1 1



, a₂ =



 2 0 1



, a₃ =



 1

−2

−1





[解]: 与 基底 直交基底 .

b₁ =a₁ =



 1 1 1



,

b2 =a2−(a₂, b₁) (b₁, b₁)b1 =



 2 0 1



−



 1 1 1



=



 1

−1 0



,

b₃ =a₃−(a₃, b₁)

(b₁, b₁)b₁− (a₃, b₂) (b₂, b₂)b₂ =



 1

−2

−1



−





−²₃

−²₃

−²₃



−





3

−2³₂

0



=





1 6 1

−6¹₃



.

直交基底 b₁, b₂, b₃ 正規化 行 正規直交基底 求 . ,求 正規直交基底c₁, c₂, c₃ ,

c₁ =

√3 3



 1 1 1



=





√3

√3 3

√3 3 3



, c₂ =

√2 2



 1

−1 0



=





√2 2

−^√₂² 0



, c₃ =√ 6





1 6 1

−6¹₃



=





√6

√6 6 6

−^√₃⁶



.

(4) a₁ =





−1 2 2



, a₂ =



 0 1 2



, a₃ =



 1 1 1





[解]: 与 基底 直交基底 .

b₁ =a₁ =





−1 2 2



,

b₂ =a₂−(a₂, b₁) (b₁, b₁)b₁ =



 0 1 2



−





−²₃

4 3 4 3



=





2

−3¹₃

2 3



,

b3 =a3−(a3, b1)

(b₁, b₁)b1− (a3, b2) (b₂, b₂)b2 =



 1 1 1



−





−¹₃

2 3 2 3



−





2

−3¹₃

2 3



=





2 3 2

−3¹₃



.

直交基底 b1, b2, b3 正規化 行 正規直交基底 求 . ,求 正規直交基底c₁, c₂, c₃ ,

c₁ = 1 3





−1 2 2



=





−¹₃

2 3 2 3



, c₂ = 1





2

−3¹₃

2 3



=





2

−3¹₃

2 3



, c₃ = 1





2 3 2

−3¹₃



=





2 3 2

−3¹₃



.



1





2





3



[解]: 与 基底 直交基底 .

b1 =a1 =



 1 2 3



,

b₂ =a₂− (a₂, b₁) (b₁, b₁)b₁ =



 2 3 4



−





10 7 20

7 30

7



=





4 7 1

−7²₇



,

b₃ =a₃− (a₃, b₁)

(b₁, b₁)b₁−(a₃, b₂) (b₂, b₂)b₂ =



 3 4 1



−



 1 2 3



−





8 3 2

−3⁴₃



=





−²₃

4

−3²₃



.

直交基底 b₁, b₂, b₃ 正規化 行 正規直交基底 求 . ,求 正規直交基底c1, c2, c3 ,

c₁ =

√14 14



 1 2 3



=





√14

√14 14 7 3√ 14 14



, c₂ =

√21 3





4 7 1

−7²₇



=





4√ 21

√21 21 21

−^2√₂₁²¹



, c₃ =

√6 4





−²₃

4

−3²₃



=





−_√^√₆⁶

6 3

−^√₆⁶



.

(6) a1 =





 1

−1 1

−1





, a2 =





 1 1 0 0





, a3 =





 0 0 1 1





, a4 =





 0 1 1 0







[解]: 与 基底 直交基底 .

b₁ =a₁ =





 1

−1 1

−1





,

b₂ =a₂− (a₂, b₁) (b₁, b₁)b₁ =





 1 1 0 0





−





 0 0 0 0





=





 1 1 0 0





,

b₃ =a₃− (a₃, b₁)

(b₁, b₁)b₁− (a₃, b₂) (b₂, b₂)b₂ =





 0 0 1 1





−





 0 0 0 0





−





 0 0 0 0





=





 0 0 1 1





,

b₄ =a₄− (a₄, b₁)

(b₁, b₁)b₁− (a₄, b₂)

(b₂, b₂)b₂− (a₄, b₃) (b₃, b₃)b₃ =





 0 1 1 0





−





 0 0 0 0





−







1 2 1 2

0 0





−





 0 0

1 2 1 2





=







−¹₂

1 2 1

−2¹₂





.

直交基底 b1, b2, b3, b4 正規化 行 正規直交基底 求 . , 求

正規直交基底 c1, c2, c3, c4 ,

c₁ = 1 2





 1

−1 1

−1





=







1

−2¹₂

1

−2¹₂





, c₂ =

√2 2





 1 1 0 0





=







√2

√2 2 2

0 0





,

c₃ =

√2 2





 0 0 1 1





=





 0

√0

2

√2 2 2





, c₄ = 1







−¹₂

1 2 1

−2¹₂





=







−¹₂

1 2 1

−2¹₂





.

(7) a1 =





 1 3 0 0





, a2 =







−1 1 0 0





, a3 =





 0 0 1 1





, a4 =





 0 0 3 4







[解]: 与 基底 直交基底 .

b₁ =a₁ =





 1 3 0 0





,

b₂ =a₂− (a₂, b₁) (b₁, b₁)b₁ =







−1 1 0 0





−







1 5 3 5

0 0





=







−⁶₅

2 5

0 0





,

b₃ =a₃− (a₃, b₁)

(b₁, b₁)b₁− (a₃, b₂) (b₂, b₂)b₂ =





 0 0 1 1





−





 0 0 0 0





−





 0 0 0 0





=





 0 0 1 1





,

b₄ =a₄− (a₄, b₁)

(b₁, b₁)b₁− (a₄, b₂)

(b₂, b₂)b₂− (a₄, b₃) (b₃, b₃)b₃ =





 0 0 3 4





−





 0 0 0 0





−





 0 0 0 0





−





 0 0

7 2 7 2





=





 0 0

−¹₂

1 2





.

直交基底 b1, b2, b3, b4 正規化 行 正規直交基底 求 . , 求 正規直交基底 c₁, c₂, c₃, c₄ ,

c₁ =

√10 10





 1 3 0 0





=







√10 10 3√

10 10

0 0





, c₂ =

√10 4







−⁶₅

2 5

0 0





=







−_√^3√₁₀¹⁰

10 10

0 0





,

c₃ =

√2





 0 0





=





 0

√0





, c₄ =√ 2





 0 0

−





=





 0 0

−^√





.

(8) a₁ =





 1 1 1 1





, a₂ =





 2 1 2 1





, a₃ =





 0

−2 1

−1





, a₄ =





 1

−3 1

−1







[解]: 与 基底 直交基底 .

b₁ =a₁ =





 1 1 1 1





,

b₂ =a₂− (a₂, b₁) (b₁, b₁)b₁ =





 2 1 2 1





−







3 2 3 2 3 2 3 2





=







1

−2¹₂

1

−2¹₂





,

b₃ =a₃− (a₃, b₁)

(b₁, b₁)b₁− (a₃, b₂) (b₂, b₂)b₂ =





 0

−2 1

−1





−







−¹₂

−¹₂

−¹₂

−¹₂





−





 1

−1 1

−1





=







−¹₂

−¹₂

1 2 1 2





,

b4 =a4− (a4, b1)

(b₁, b₁)b1− (a4, b2)

(b₂, b₂)b2− (a4, b3) (b₃, b₃)b3 =





 1

−3 1

−1





−







−¹₂

−¹₂

−¹₂

−¹₂





−







3

−2³₂

3

−2³₂





−







−¹₂

−¹₂

1 2 1 2





=







1

−2¹₂

−¹₂

1 2





.

直交基底 b₁, b₂, b₃, b₄ 正規化 行 正規直交基底 求 . , 求 正規直交基底 c₁, c₂, c₃, c₄ ,

c₁ = 1 2





 1 1 1 1





=







1 2 1 2 1 2 1 2





, c₂ = 1







1

−2¹₂

1

−2¹₂





=







1

−2¹₂

1

−2¹₂





,

c₃ = 1







−¹₂

−¹₂

1 2 1 2





=







−¹₂

−¹₂

1 2 1 2





, c₄ = 1







1

−2¹₂

−¹₂

1 2





=







1

−2¹₂

−¹₂

1 2





.

(9) a1 =





 1 1

−1

−1





, a2 =





 1 1 1

−1





, a3 =





 1

−1 1 1





, a4 =





 1 1 1 1







[解]: 与 基底 直交基底 .

b₁ =a₁ =





 1 1

−1

−1





,

b₂ =a₂− (a₂, b₁) (b₁, b₁)b₁ =





 1 1 1

−1





−







1 2 1

−2¹₂

−¹₂





=







1 2 1 2 3

−2¹₂





,

b₃ =a₃− (a₃, b₁)

(b₁, b₁)b₁− (a₃, b₂) (b₂, b₂)b₂ =





 1

−1 1 1





−







−¹₂

−¹₂

1 2 1 2





−







1 6 1 6 1

−2¹₆





=







4

−3²₃

0

2 3





,

b₄ =a₄− (a₄, b₁)

(b₁, b₁)b₁− (a₄, b₂)

(b₂, b₂)b₂− (a₄, b₃) (b₃, b₃)b₃ =





 1 1 1 1





−





 0 0 0 0





−







1 3 1 3

1

−¹₃





−







2

−3¹₃

0

1 3





=





 0 1 0 1





.

直交基底 b1, b2, b3, b4 正規化 行 正規直交基底 求 . , 求 正規直交基底 c₁, c₂, c₃, c₄ ,

c₁ = 1 2





 1 1

−1

−1





=







1 2 1

−2¹₂

−¹₂





, c₂ =

√3 3







1 2 1 2 3

−2¹₂





=







√3

√6 3

√6 3 2

−^√₆³





,

c₃ =

√6 4







4

−3²₃

0

2 3





=







√6 3

−^√₆⁶

√0

6 6





, c₄ =

√2 2





 0 1 0 1





=







√0

2 2

√0

2 2





.