? ???A,B๊ฐ ๋ชจ๋ i,j? ????aij=bij๋ฅ?๋ง์กฑํ๋ฉ??๋ก ๊ฐ??equal)๊ณ ?๊ณ A=B ๋ก ???ธ๋?
? ???A,B๊ฐ ๋ชจ๋ i,j? ????aij=bij๋ฅ?๋ง์กฑํ๋ฉ??๋ก ๊ฐ??equal)๊ณ ?๊ณ A=B ๋ก ???ธ๋?
? ???A=[aij]m?n , B=[bij]m?n ? ?ค์ k ? ????
A? B? ??sum) A+B ? A? ?ค์นผ?ผ๋ฐฐ(scalar multiple) kA ๋ฅ??ค์๊ณ?๊ฐ???????
A+B=[aij + bij ]m?n , kA=[kaij]m?n
? ???A=[aij]m?p , B=[bij]p?n ? ????A? B? ๊ณ?product) AB๋ฅ??ค์๊ณ?๊ฐ???????
AB=[cij]m?n ?ฌ๊ธฐ?,cij = ai1b1j + ai2b2j + ??+aipbpj= ? aik bkj (1 ??i ??m , 1 ??j ??)
A๊ฐ n์ฐจ์ ??ฌ๊???ฌ์??, A? ๊ฑฐ๋ญ?๊ณฑ์ ?ค์๊ณ?๊ฐ???????
A0 = In , Ak = A ... A (k ๊ฐ)
???A=[aij ]m?n ? ????A? ?์น???transpose of A)? AT๋ก ???ธ๋?
??ฌ๊????A๊ฐ AT = A ๋ฅ?๋ง์กฑํ๋ฉ?A๋ฅ??์นญ???symmetric matrix)?ด๋??๊ณ ,
AT =-A ๋ฅ?๋ง์กฑํ๋ฉ?๋ฐ?์นญ???skew symmetric matrix)?ด๋ผ๊? ???
mxn ???E ๊ฐ ?ค์ ?ฑ์?? ๋ง์กฑํ ?, ? ?ฌ๋ค๋ฆฌ๊ผ?row echelon form)?ด๋ผ๊? ???
(i) ?ฑ๋???๋ชจ๋ 0?????์กด์ฌํ๋ฉ?๊ท??? ??ฌ์ ๋ง???? ?์น???
(ii) ๊ฐ ??? ์ฒ??ผ๋? ???? 0???? ?ฑ๋?? 1?ด๋? ?ด๋, ??1? ๊ท??? ? ? ?ฑ๋?(leading entry)?ด๋ผ๊? ???
(iii) i ?๊ณ?i+1 ? ๋ชจ๋? ? ??ฑ๋???์กด์ฌํ๋ฉ?(i+1) ?? ? ??ฑ๋?? i ?? ? ? ?ฑ๋?๋ณด๋??ค๋ฅธ์ชฝ์ ?์น???
?, ???E ๊ฐ ??ฌ๋ค๋ฆฌ๊ผด์ด๊? ?ค์ ?ฑ์?? ๋ง์กฑํ๋ฉ?E๋ฅ?๊ธฐ์?? ?ฌ๋ค๋ฆฌ๊ผ?(reduced row echelon form)?ด๋ผ๊? ???
(iv) ?ด๋??? ? ??ฑ๋?? ?ฌํจํ? ?ด์ ?ค๋ฅธ ?ฑ๋?? ๋ชจ๋ 0?ด๋?
* ??ผ๋? ? ?ฌ๋ค๋ฆฌ๊ผด์ ๊ฐ?จํ REF๋ก, ๊ธฐ์?? ?ฌ๋ค๋ฆฌ๊ผด์ RREF๋ก ???ด๊ธฐ๋ก ???
mxn ???A? ๊ด? ?ค์ ?ฐ์ฐ์ ๊ธฐ๋ณธ??ฐ์?elementary row operation)?ด๋ผ๊? ???
E1 : A? ? ? i ?๊ณ?j ?? ?๋ก ๋ฐ๊พผ๋?
E2 : A? i ?? 0 ???? ?? k ๋ฅ?๊ณฑํ??
E3 : A? i ?? k ๋ฐฐํ??j ?? ????
??ผ๋? ๊ธฐ๋ณธ??ฐ์ฐ์ ?ค์๊ณ?๊ฐ??๊ธฐํธ๋? ???ด๊ธฐ๋ก ???
( [?? 1] ์ฐธ์กฐ )
E1 : Ri <--> Rj
E2 : k Ri
E3 : k Ri + Rj
???A? ๊ธฐ๋ณธ??ฐ์ฐ์ ??????ป์ด์?? ??ฌ์ B???๋ฉ?A? B? ??์น(row equivalent)?ผ๊? ???
n์ฐจ์ ??ฌ๊????A? ?????ค์? ๋ง์กฑํ? ???B๊ฐ ์กด์ฌํ๋ฉ?A? ๊ฐ?ญ(invertible)?ด๋ผ๊? ???
AB = In = BA
?ด๋, B๋ฅ?A? ?ญ???inverse matrix)?ด๋ผ๊? ?๋ฉ? ?ด๋ฌํ B๊ฐ ์กด์ฌํ์ง ??ผ๋ฉด A? ๋น๊ฐ?ญ(noninvertible)?ด๋ผ๊? ???
??ฐ์? ์ง??S={1,2,.........n}? ์น?(permutation, ????ด๋ S?? S๋ก? ?ผ๋???? ?จ์ ??ด๋? ?จ์
? ๊ฐ?จํ (i1, i2, ... ,in)?ผ๋? ?ฐ๊ธฐ๋ก ???
[ ์ฃผ์: ?ด๊?? ์น?๊ตฐ์? ?? ?ฐ๋ ??์น? ๋ถ?ธ์? ?ค๋ฅธ?๋ฏธ์ด๋?
?จ์ sgn:Sn ? {-1, +1}? ?ค์๊ณ?๊ฐ???????
???A=[aij]๊ฐ n์ฐจ์ ??ฌ๊???ฌ์??, A???ฌ์(determinant)? det(A) ?? |A|๋ก ???ด๊? ?ค์๊ณ?๊ฐ???????
??ผ๊????/b>(upper triangular matrix), ??ผ๊????/b>(lower triangular matrix),?ผ๊????/b>(triangular matrix) ์ฃผ๋๊ฐ? ?ฑ๋? ??? ?ฑ๋???๋ชจ๋ 0????ฌ๊???ฌ์ ??ผ๊????/b>(upper triangular matrix), ?์ชฝ์ ?ฑ๋???๋ชจ๋ 0????ฌ๊???ฌ์ ??ผ๊????/b>(lower triangular matrix)?ด๋ผ๊? ??? ??ผ๊???ฌ๊ณผ ??ผ๊???ฌ์ ?ตํ??b> ?ผ๊????/b>(triangular matrix)?ด๋ผ๊? ???
??ฌ๊????A=[aij]? i?๊ณ?j?ด์ ?๊ฑฐํ??๋ง? ๋ถ๋ถ??ฌ์ A(i|j)???๊ณ ๊ท?? ??ฌ์ |A(i|j)|๋ฅ? aij? ???ฌ์(minor)?ด๋???? ?,
Aij = (-1)i+j | A (i|j) |
Aij ? ?ฌ์ธ์(cofactor)?ผ๊? ???
n์ฐจ์ ??ฌ๊????A=[aij]? ?ฑ๋? aij? ?ฌ์ธ์๋ฅ?Aij??? ?, ???[Aij]T๋ฅ?A? ?๋ฐ???adjoint matrix)?ด๋??๊ณ , adjA๋ก ???ธ๋? ์ฆ,
?ฐ๋ฆฝ?ผ์ฐจ๋ฐฉ์ ?? ?ด์ ?? ๊ณต์? ๋ง?ค๋ฉด ?ค์ ? ๊ณ?ฐ์ ๋ณต์กํ??ผ๋ ?ด์ ?ฑ์?? ์กฐ์ฌํ ?? ๋งค์?? ?ฉํ?? ?ค์? n๊ฐ? ๋ฏธ์??๋ฅ? ๊ฐ์ง? n๊ฐ? ?ผ์ฐจ๋ฐฉ์ ??ผ๋? ?ด๋ฃจ?ด์? ?ฐ๋ฆฝ๋ฐฉ์ ?? ?ด๋? ๊ตฌํ? ๊ณต์? ?๊ฐ???
??๊ณต์? ?ฌ๋๋จ?๊ณต์(Cramer's rule)???ผ์ด๋??๋ฉ? ???ฐ๋ฆฝ๋ฐฉ์ ?? AX=B๋ก ????? ??? ?ด๋,|A|? 0?ด๋ฉด ????๋ฆฝ๋ฐฉ???
? ?ผํ ??/p>
? ๊ฐ??? ?ฌ๊ธฐ? Mj (j=1, 2,........,n)? A? j?ด์ B๋ก ๋ฐ๊พ???ฌ์ด๋?
? ๋ฒกํ?x, y? ?ค์นผ??k? ????? ๋ฒกํฐ์ ??x+y? k? ?? x? ?ค์นผ?ผ๋ฐฐ kx๋ฅ??ค์๊ณ?๊ฐ???????
(i) x+y? x, y? ????๊ฒฐ์ ?? ???ฌ๋??? ?๊ฐ? ?ผ๋? ???? ๋ฒกํฐ์ด๋?/p>
(๊ทธ๋ฆผ 3.2 (a)).
(ii) kx? k>0 ?ด๋ฉด x? ๋ฐฉํฅ์?๊ฐ?ผ๋ฉด? ๊ธธ์ด๋ k๋ฐ?????ป์ด์? ? ๋ฒกํฐ์ด๊?, k<0 ?ด๋ฉด x? ๋ฐฉํฅ์?๋ฐ??ด๋ฉด? ๊ธธ์ด๋ |k| ๋ฐ?? ???ป์ด์?? ๋ฒกํฐ์ด๋?(๊ทธ๋ฆผ 3.2 (b)). ?? k ๊ฐ 0?ด๋ฉด kx? ๊ธธ์ด๊? 0 ??๋ฒกํฐ์ด๋?
๊ทธ๋ฆผ 3.2
???ค์?ค์ ??์ก?(x1, x2, x3)๋ฅ?๊ณต๊?๋ฒกํ?/b>(vector in space)???๊ณ
๋ก ???ธ๋? ?ด๋ ?ค์ x1, x2, x3๋ฅ?
R3? ๋ฒกํ?x=(x1, x2, x3), y=(y1,y2,y3)? ?ค์นผ??k? ???? ? ๋ฒกํฐ์ ??vector sum) x+y?, k? ?? x? ?ค์นผ?ผ๋ฐฐ(scalar multiple) kx๋ฅ??ค์๊ณ? ๊ฐ???????
(i) x+y=(x1+y1, x2+y2, x3+y3)
(ii) kx=(kx1, kx2, kx3)
R3? ? ๋ฒกํ?x=(x1, x2, x3), y=(y1,y2,y3)? ?????ค์
x1y1 + x2y2 + x3y3
๋ฅ?x? y? ?ด์ (inner product ?? scalar product) ?ด๋??๊ณ x·y๋ก ???ธ๋? ์ฆ,
x·y = x1y1 + x2y2 + x3y3
R3? ๋ฒกํ?x=(x1, x2, x3), y=(y1,y2,y3)? ????x, y? ?ธ์ (cross product ?? vector product)? x×y๋ก ???ด๋ฉฐ ?ค์๊ณ?๊ฐ???????
x×y=(x2y3-x3y2)i+(x3y1-x1y3)j+(x1y2-x2y1)k
n๊ฐ? ?ค์? ??์ก?(x1, x2, ... ,xn)? n-์ฐจ์๋ฒกํ?n-dimensionalvector)?? ?๊ณ
?ผ๋? ???ธ๋? ?ด๋ ?ค์ x1, x2, ... ,xn? x? ?ฑ๋??ด๋? ???
๋ชจ๋ n-์ฐจ์๋ฒกํ??์ฒด์ ์ง?ฉ์ Rn?ผ๋? ???ธ๋? ์ฆ,Rn={(x1, x2, ... ,xn)|xi?R, i=1, 2,.......,n}
?ผ๋? ?ผ๋ ค๊? ?? ?? Rn? ??๋ฅ?๊ฐ?จํ ๋ฒกํฐ๋???? ?นํ n=1???, R1? ?ค์ ?์ฒด์ ์ง?ฉ๊ณผ ?ผ์??๋ฏ๋ก R1? ๊ฐ?จํ R๋ก ???ธ๋?
Rn? ๋ฒกํ?/p>
? ????xi=yi(i=1,2,.......,n)?ด๋ฉด x=y?ผ๊? ???
Rn? ๋ฒกํ?/p>
? ?ค์นผ??k? ???? ? ๋ฒกํฐ์ ??x+y? k? ?? x? ?ค์นผ??๋ฐ?kx๋ฅ?๊ฐ๊ฐ ?ค์๊ณ?๊ฐ???????
??, Rn ?? ๋ชจ๋ ?ฑ๋???0??๋ฒกํฐ๋? ?๋ฒกํฐ๋??๊ณ 0?ผ๋? ???ธ๋? ๊ทธ๋ฌ๋ฉด ??? ๋ฒกํ?x?R? ???? x+0=x, x+(-1)x=0
???ฑ๋ฆฝ?จ์ ?ฝ๊? ? ? ??? ?ฌ๊ธฐ? (-1)x=-x๋ก ???๋ฉ?-x๋ฅ?x? ?๋ฒกํฐ๋????
Rn? ๋ฒกํ?=(x1, x2, ... ,xn)? ????
||x||=sqrt(x12, x22, ... ,xn2)
? x? ?ฌ๊ธฐ(norm)?????
?? ???? ||x||? ???? ? P(x1, x2, ... ,xn)? ?ด๋ฅด? ๊ฑฐ๋ฆฌ๋ก ???จ์ ?๋ฏธํ??
?ฐ๋ผ์ R^n? ? ๋ฒกํ?x=(x1, x2, ... ,xn) y=(y1, y2, ... ,yn)? ????|x-y||? ? ? P(x1, x2, ... ,xn)? Q(y1, y2, ... ,yn) ?ฌ์ด์ ๊ฑฐ๋ฆฌ๋ก ?? ??? ์ฆ,
||x-y||=sqrt((x1-y1)2+ ... +(xn-yn)2)
Rn? ๋ฒกํ?x=(x1, x2, ... ,xn), y=(y1, y2, ... ,yn)? ?????ค์ x1y1+x2y2+ ... +xnyn ? x? y? ?ด์ ?ด๋??๊ณ x·y๋ก ???ธ๋? ์ฆ, x·y=x1y1+x2y2+ ... +xnyn
?ด์ ? ??๋ก ๋ถ??Rn? ๋ฒกํ?x? ?????ค์? ?ป๋??
x·x=||x||2
Rn? ๋ฒกํ?x, y? ????x·y=||x||||y||cosฮธ,0?คฮธโคฯ ??ฮธ๋ฅ?x? y๊ฐ ?ด๋ฃจ? ๊ฐ(angle)?ด๋????
V? ์ง?ฉ์ด๊?, V?? ๋ฒกํฐํ?vector addition)(?? ๋ฒกํ??ง์ )?ด๋ผ๊? ๋ถ๋ฅด๋ ?ฐ์?์ฆ, V? ? ?? u, v? V? ?? u+v๋ฅ?????ค๋ ?ฐ์ฐ๊ณผ ?ค์นผ?ผ์ ?? ๊ณ?multiplication by scalars)?ด๋ผ๊? ๋ถ๋ฅด๋ ?ฐ์?์ฆ, V? ?? u? ?(?ด๋? ๋ณดํ??ค์นผ??scalar)?ผ๊? ๋ถ๋ฆ) a? V? ?? au๋ฅ?????ค๋ ?ฐ์ฐ์????????ค๊? ??. ??? ๊ฐ์ง ?ฐ์ฐ์? ?ค์ 8๊ฐ์ง ์กฐ๊ฑด? ๋ง์กฑํ ? V๋ฅ?๋ฒกํฐ๊ณต๊ฐ(vector space)?ด๋??๊ณ , V? ??๋ฅ?๋ฒกํฐ๋ผ๊? ???
(i) V? ๋ชจ๋ ?? u, v? ????u + v = v + u.
(ii) V? ๋ชจ๋ ?? u, v, w? ????(u + v) + w = u + (v + w).
(iii) V? ๋ชจ๋ ?? u? ????u + O = u = O + u ๋ฅ?๋ง์กฑํ? O๊ฐ V?? ์กด์ฌํ??
(iv) V? ๊ฐ ?? u? ????V? ?? u'??์กด์ฌํ??u + u' = O = u' + u ๊ฐ ?ฑ๋ฆฝ???
(v) ๋ชจ๋ ?ค์นผ??a, b? V? ๋ชจ๋ ?? u? ????(a + b)u = au + bu.
(vi) ๋ชจ๋ ?ค์นผ??a? V? ๋ชจ๋ ?? u, v? ????a(u + v) = au + av.
(vii) ๋ชจ๋ ?ค์นผ??a, b? V? ๋ชจ๋ ?? u? ????(ab)u = a(bu).
(viii) V? ๋ชจ๋ ?? u? ????1u = u.
?? 1? ์กฐ๊ฑด (iii)?? O๋ฅ?V? ?๋ฒกํ?zero vector)?ผ๊? ??? ? ์กฐ๊ฑด (iv)?? u'? u? ?ง์ ? ๊ด? ?ญ??ด๋ผ๊? ?๋ฉ? ๋ณดํ?๊ธฐํ?-u๋ก ???ธ๋? ์กฐ๊ฑด (i), ์กฐ๊ฑด (ii)๋ฅ?๊ฐ๊ฐ ๋ฒกํ??ง์ ? ๊ต?๋ฒ์น(commutative law), ๋ฒกํ??ง์ ? ๊ฒฐํฉ๋?์น(associative law)?ด๋ผ๊? ??? ?นํ (ii)? ????๋ฒกํฐ๊ณต๊ฐ?? ??๋ฒกํ?u, v, w? ?ฉ์ ๊ด?????u + v + w๋ก ???ด๊ธฐ? ???
v1, ... , vn??W๊ฐ V? ๋ถ๋ถ์ง?ฉ์ด๋???. ?ค์ ??์กฐ๊ฑด???ฑ๋ฆฝ? ? W๋ฅ?V? ๋ถ๋ถ๊ณต๊?(subspace)?ด๋ผ๊? ???
(i) V? ?๋ฒกํ?O๊ฐ W? ????
(ii) W? ๋ชจ๋ ?? u, v? ????u+v๊ฐ W? ????
(iii) ๋ชจ๋ ?ค์นผ??a? W? ๋ชจ๋ ?? u? ????au๊ฐ W? ????
?? 2?? ์กฐ๊ฑด (ii)๋ฅ?๋ณดํ?W? ๋ฒกํ??ง์ ? ๊ด???ซํ ??ค๊? ?๋ฉ? ์กฐ๊ฑด (iii)? ๋ณดํ?W? ?ค์นผ?ผ์ ?? ๊ณฑ์ ? ๊ด?? ?ซํ ??ค๊? ???/p>
v1, ... , vn??V? ๋ฒกํฐ๋ค์ด๊?, a1, ... , an?? ?ค์นผ?ผ๋ค์?? ??a1v1 + ... + anvn? ๋ฒกํ? v1, ... , vn? ?ผ์ฐจ๊ฒฐํ?/b>(linear combination)?ด๋ผ๊? ???
๋ฒกํ?v1, ... , vn? a1v1 + ... + anvn๊ฐ ? ๊ฐ?์ง? ๊ฒฝ์ฐ๊? ?ค์? a1 = ... = an = 0 ??๊ฒฝ์?๋ฟ??? ๋ฒกํ?v1, ... , vn? ?ผ์ฐจ? ๋ฆ?/a>(linearly independent)?ด๋ผ๊? ??? ๊ทธ๋ ์ง ?? ? ์ฆ, ?ค์นผ??a1, ... , an์ค? 0?? ?? ๊ฒ?????ฐ๋ ๋ถ๊ตฌํ๊ณ ?ผ์ฐจ๊ฒฐํ?a1v1 + ... + anvn๊ฐ ?๋ฒกํฐ์ ๊ฐ?์ง? ๊ฒฝ์ฐ๊? ?๊ธ?? ๋ฒกํ?v1, ... , vn? ?ผ์ฐจ์ข ?(linearly dependent)?ด๋ผ๊? ???
W๊ฐ V? ?ด๋ผ๊? ??. W?? ??? ๋ฒกํ?u๊ฐ ๋ฒกํ?v1, ... , vn? ?ผ๋? ???๊ณ , ? ?ญ?ผ๋? ๋ฒกํ?v1, ... , vn? ?ผ๋? ???? ๋ชจ๋ ๋ฒกํฐ๊? W? ?? ? ๋ฒกํ?v1, ... , vn? W๋ฅ???ฑํ??span)๊ณ ???
V?? ๋ฒกํ?v1, ... , vn???ค์ ? ๊ฐ์ง ์กฐ๊ฑด? ๋ง์กฑํ ? ์ง??{v1, ... , vn}? V? ๊ธฐ์ (basis)?ผ๊? ???
(i) ๋ฒกํ?v1, ... , vn? ?ด๋?
(ii) ๋ฒกํ?v1, ... , vn? V๋ฅ????
V?? ๊ฐ n๊ฐ? ๋ฒกํฐ๋? ?ด๋ฃจ?ด์ ??? ? V? ์ฐจ์(dimension)? n?ด๋??๊ณ , ๊ธฐํ?dim V = n ?ผ๋? ???ธ๋? ๋ชจ๋ ??ฐ์ k? ????/p>
V??? ??๋ฒกํ?v1, ... , vk ๊ฐ ์กด์ฌํ ? V๋ฅ?๋ฌดํ์ฐจ์ ๋ฒกํฐ๊ณต๊ฐ?ด๋? ?๊ณ , ๊ธฐํ?dim V = ? ๋ก ???ธ๋?
๋ง??V? ? ๊ฐ ?๋ก ?ค๋ฅธ ๊ฐ?? ๋ฒกํฐ๋ค๋? ?ด๋ฃจ?ด์ ???ค๋ฉด, ?? 7??? ??? ? ๋ชป๋ ๊ฒ?ด๋? ๊ทธ๋ฌ๋ ? ????? ๊ธฐ์ ? ๊ฐ? ๊ฐ?? ๋ฒกํฐ๋ค๋? ?ด๋ฃจ?ด์ ???ผ๋ฏ๋ก ?? 7?? ? ??? ? ??? ๊ฒ?ด๋?
์ง??B = {v1, ... , vn} ??V? ?ผ๊? ??. V?? ๋ฒกํ?v๊ฐ ?ผ์ฐจ๊ฒฐํ?c1v1 + ... + cnvn ?ผ๋? ??? ? ?ค์นผ??c1, ... , cn ?ค์ n-tuple (c1, ... , cn) ? ๋ฒกํ?v? ์ฃผ์ด์? ๊ธฐ์ B? ๊ด? ์ข??ผ๊? ???
V, W๊ฐ ๋ฒกํฐ๊ณต๊ฐ?ด๊? L:V?W??? ?๋ณ????, L? ?? ???0???? V? ๋ฒกํ??์ฒด์ ์ง?ฉ์ L? ??/b>(kernel)?ด๋? ?๊ณ kerL๋ก ???ธ๋?
??ฌ๊????A, B? ?????ค์? ๋ง์กฑํ? ๊ฐ?ญ???P๊ฐ ์กด์ฌํ ? B? A? ?ฎ?(similar)??ฌ์ด๋ผ๊? ???
B=P-1AP
ฮป๊ฐ n์ฐจ์ ??ฌ๊????A? ๊ณ ? ๊ฐ???, ?์ฐจ์ฐ๋ฆฝ๋ฐฉ์ ? (ฮปIn -A)x=0? ?ด๊ณต๊ฐ? ฮป? ???? A? ๊ณ ? ๊ณต๊?(eigenspace)?ด๋ผ๊? ???
? ??? ????ฮป? ???? A? ๊ณ ? ๊ณต๊?? ฮป? ???? A? ๊ณ ? ๋ฒกํ??์ฒด์ ?๋ฒกํฐ๋? ?ด๋ฃจ?ด์? ์ง?ฉ์ด๋ฉฐ, ?ด๋ ๋ฒกํฐ๊ณต๊ฐ Rn? ๋ถ๋ถ๊ณต๊??ด๋?
A๊ฐ ?ด๋??๊ฐ??ฌ๊ณผ ?ฎ???ฌ์?? ์ฆ, ??นํ ๊ฐ?ญ???P๊ฐ ์กด์ฌํ??P-1AP ๊ฐ ?๊ฐ??ฌ์?? A๋ฅ??๊ฐ?๊ฐ?ฅํ(diagonalizable) ??ฌ์ด๋??๋ฉ????P? A ๋ฅ??๊ฐ??? ??ฌ์ด๋ผ๊? ???
??ฌ๊????A? ????A-1=AT?ด๋ฉด A๋ฅ?b> ์ง๊ต???/b>(orthogonal matrix)?ด๋ผ๊? ???
? ??๋ก๋ถ???ค์? ?๋ก ?์น?? ?ฝ๊? ? ? ??? ?ด๊?? ์ฆ๋ช ? ?ฐ์ต๋ฌธ?๋ก ?จ๊ธด??
(1) A๊ฐ ์ง๊ต??ฌ์ด๋?
(2) ATA=In=AAT
(3) A? ???)๋ฒกํฐ๋ค์ ?๊ท์ง๊ต์ง?ฉ์ ?ด๋ฃฌ??
??ฌ๊????A? ????A๋ฅ??๊ฐ??? ์ง๊ต???P๊ฐ ์กด์ฌํ ? ์ฆ, P-1AP=D ?? PTAP=D?? ์ง๊ต???P? ?๊ฐ???D๊ฐ ์กด์ฌํ ? A? ์ง๊ต?๊ฐ?๊ฐ??/b>(orthogonally diagonalizable)??ค๊? ?๋ฉ? P? A๋ฅ?์ง๊ต?๊ฐ??? ??ฌ์ด๋????
?? ???A๋ฅ??คํญ?? ?๋ฐ???(Companion matrix) ?ด๋???? ???A? ?น์?๋ฐฉ์ ?? ๋ฐ๋ก p(x)?ด๋? ?, ?? ?คํญ? ๊ฐ??์ต๊ณ ์ฐจํญ? ๊ณ?๊ฐ 1???คํญ?? ๋ชจ๋(monic)?คํญ??ด๋ผ๊? ???
? ? ?ค์ a, b? i=sqrt(-1)? ????z=a+bi๋ฅ?๋ณต์?(complex number)?????
?? ๊ฐ????? ๋ณต์? z=a+bi? ????a๋ฅ???๋ณต์?? ?ค์๋ถ๋ถ, b๋ฅ???๋ถ๋ถ?ด๋??๋ฉ?๋ณต์? ?์ฒด์ ์ง?ฉ์ C๋ก ???ธ๋? ์ฆ,
C= {a+bi: a, b?R} ๋ณต์? z=a+bi?? b=0?ด๋ฉด z=a+0i?ด๋ฏ๋ก ??? ?ค์? ??๋ถ๋ถ??0??๋ณต์??ด๋? ??, a=0?ด๋ฉด z=0+bi=bi?ด๋? ?ด์๊ฐ???ค์๋ถ๋ถ??0??๋ณต์?๋ฅ????(pure imaginary number)?????
? ๋ณต์? z1=a1+b1i, z2=a2+b2i ? ????a1=b1, a2=b2?ด๋ฉด,z1=z2 ???
๋ณต์? z=a+bi? ????๋ณต์? a-bi๋ฅ?z? ์ผค๋ ๋ณต์?(conjugate)???๊ณ z-๋ก ???ธ๋? ์ฆ,z-=a-bi ??, ๋ณต์? z=a+bi? ??๊ฐ(modulus) |z|๋ฅ??ค์๊ณ?๊ฐ???????
|z|=sqrt(a2+b2)
Cn? ? ๋ฒกํ?u=(u1, u2, ... ,un), v=(v1, v2, ... ,vn)? ? ?ด๋ฆฌ? ?ด์ u·v, u? ? ?ด๋ฆฌ? ?ธ๋? ||u||, u ? v ?ฌ์ด์ ? ?ด๋ฆฌ?๊ฑฐ๋ฆฌ d(u, v)๋ฅ?๊ฐ๊ฐ ?ค์๊ณ?๊ฐ???????
๋ณต์๋ฒกํฐ๊ณต๊ฐ V? ??? ๋ฒกํ?, v, w? ?ค์นผ??c?C? ?????ค์ ์กฐ๊ฑด? ๋ง์กฑํ? V×V?? C๋ก? ?จ์ < , >๋ฅ?V? ?ด์ (?? Hermitian ?ด์ )?ด๋? ???
?ด์ ? ๊ฐ? ๋ณต์๋ฒกํฐ๊ณต๊ฐ? ๋ณต์?ด์ ๊ณต๊?(complex inner product space)??? ??๋ฆฌ๊ณต๊ฐ(unitary space)?ด๋???? ??, ??? ๋ณต์๋ฒกํ?u, v? ???? <u, v>=0?ด๋ฉด u ? v? ์ง๊ต???orthogonal)๊ณ ???
?ด์ ๊ณต๊?? ??๋ก ๋ถ???ค์ ?ฑ์?? ๋ฐ๋ก ?ป์ ? ???
๋ณต์?ด์ ๊ณต๊? V?? ๋ฒกํ?u? ?ธ๋?(norm)๊ณ? V?? ๋ฒกํ?u, v?ฌ์ด์ ๊ฑฐ๋ฆฌ(distance)๋ฅ? ๊ฐ๊ฐ ?ค์๊ณ?๊ฐ?ด์ ????
* Cn ??? Cauchy-Schwarz ๋ถ?ฑ์, ?ผ๊?๋ถ?ฑ์ ? ?ด๋ฆฌ? ?ด์ ????????? ๋ณต์?ด์ ๊ณต๊? Cn? ? ๋ฒกํ?/p>
?ด๋? ?ด๊?? ๊ฐ๊ฐ Cn ??? Cauchy-Schwarz ๋ถ?ฑ์, ?ผ๊?๋ถ?ฑ์?ด๋ผ๊? ???
๋ณต์???A ๊ฐ A=A* ?ด๋ฉด, A ๋ฅ?b> Hermitian???/b>?ด๋????
๋ณต์???A ๊ฐ A=-A* ?ด๋ฉด A๋ฅ?b> ๋ฐ-Hermitian(skew-Hermitian) ???/b>?ด๋????
???U?Mn(C)๊ฐ U*U=In?ด๋ฉด U๋ฅ?? ??๋ฆ?unitary)???/b>?ด๋ผ๊? ???
??? ????U๊ฐ ? ??๋ฆฌํ?ฌ์ด๋ฉด U*=U-1?ด๋? ??, U ? j๋ฒ์ง??ด๋ฒก?ฐ๋?uj?? ?๋ฉ?/p>
?ด๋ฏ๋ก U๊ฐ ? ??๋ฆฌํ?ฌ์???์ถฉ๋?์กฐ๊ฑด? U? ?ด๋ค์??๊ท์ง๊ต์ง??/b>? ?ด๋ฃฌ??
๋ชจ๋ ??ฌ์??๊ฐ?๊ฐ?ฅํ ๊ฒ? ??์ง๋ง Schur?๋ฆฌ๋? ?ด์ฉํ๋ฉ??๊ฐ?๊ฐ?ฅํ์ง ?? ???A? ?๊ฐ??ฌ๊ณผ ? ?ฌํ ???JA? ?ฎ?(similar)???๊ฒ ? ? ??? ?ด๋ฌํ ???JA๋ฅ?A? Jordan ?์ค??ด๋???? Jordan?์ค?? ?ด์ฉํ๋ฉ?๋ชจ๋ ??ฌ์ ?๊ฐ??ฌ๊ณผ ? ?ฌํ ???JA๋ก ๋ฐ๊พธ์????A? ๊ด? ๊ณ?ฐ๋? ?ด๋?? ?๊ฐ ? ? ??? ?ด๋ฌํ Jordan ?์ค?? ๊ตฌํ? ๋ฐฉ๋? ?ฑ์ 7??5?? ์ฐธ์กฐ?๊ธ?๋ฐ???
? ?ด์ฐจ๋ฐฉ์ ? (1)? ๊ด? ?ด์ฐจ???ด๋ผ๊? ???
A=[aij]๊ฐ n์ฐจ์ ?ค๋์นญ??ฌ์ด๊?, n๊ฐ? ๋ณ? x1,x2 , ...
, xn ? ?ฑ๋??ผ๋? ๊ฐ?
Rn?? ?ด์ฐจ???ด๋ผํ??
?๊ณ?(critical point) : f? ?ผ๊??ธ๋?จ์? a?? ๋ชจ๋ 0.
f? Hessian???:x=a?? f? ?ด๊? ?ธ๋?จ์๋ฅ??ฑ๋??ผ๋? ๊ฐ์ง? ?ค์๊ณ?๊ฐ? n์ฐจ์ ?์นญ???H
x=a ?? ?จ์ f? ?ด์ฐจ??? q(x)=xTAx ๋ก ????? ?ฌ๊ธฐ? H? (3)๊ณ?๊ฐ? Hessian ??ฌ์ด๋?
?๋ฆ?8.6 ?จ์ f:Rn ?R๊ฐ ?๊ณ? x=a?? ?ฐ์??3๊ณ?ธ๋?จ์๋ฅ?๊ฐ๊ณ , ????? f? ?ด์ฐจ??? q(x)=xTHx ?ผ๊? ? ?, ?ค์???ฑ๋ฆฝ???
(1) q๊ฐ ?? ?๋ถ?ธ์ด๋ฉด f(a)? ๊ทน์๊ฐ?ด๋?
(2) q๊ฐ ?? ?๋ถ?ธ์ด๋ฉด f(a)? ๊ทน๋๊ฐ?ด๋?
(3) q๊ฐ ๋ถ?๋ถ?ธ์ด๋ฉด, f(a)? ๊ทน๋๊ฐ? ??๊ณ ๊ทน์๊ฐ? ????
?ด๋, x=a ๋ฅ???ฅ์ (saddle point)?ด๋ผ๊? ???
?๋ฆ?8.6 ?? n=2 ?ผ๋ f ๊ฐ ?ด๋???จ์๊ฐ ?๊ณ f? Hessian???/b>??(4)? ๊ฐ?ผ๋ฏ๋ก ?๋ฆ?8.5 ? 8.6 ? ????๋ฏธ์ ๋ถ??? ๋ฐฐ์??ด๋???จ์? ๊ทน๋๊ทน์??๋ฒ? ?ป๋??