4 Da.ng to`an phu.o.ng
4.2.4Da.ng to`an phu.o.ng x´ac d¯i.nh ˆam, x´ac d¯i.nh du.o.ng, luˆa.t
t´ınh.
D- i.nh nghı˜a 4.5. Da.ng toa`n phu.o.ng ω trˆen R− khˆong gian vector E d¯u.o.. c go.i la` xa´c d¯i.nh du.o.ng (hay xa´c d¯i.nh ˆam) nˆe´u ∀x ∈ E, x 6= 0 ta co´ : ω(x) >0 (hay ω(x) < 0).
4.2. Da.ng to`an phu.o.ng. 77 D- i.nh ly´ 4.2. Nˆe´u ω la` mˆo.t da.ng toa`n phu.o.ng xa´c d¯i.nh du.o.ng (ˆam) trˆen R−
khˆong gian vector E thı` tˆ` n tai mˆo.t co. so.’ cu’ao E sao cho ω co´ da.ng chı´nh t˘a´c sau:
ω(x) = k1x21+k2x22+· · ·+knx2n, v´o.i ki >0 (ki <0), ∀i = 1, n.
D- i.nh ly´ 4.3.
(i) Da.ng toa`n phu.o.ng ω trˆen R− khˆong gian vector E la` xa´ c d¯i.nh du.o.ng khi va` chı’ khi tˆ` n ta.i mˆo.t co. so.’ cu’ao E sao trong co. so.’ ˆa´y, ca´ c d¯i.nh th´u.c con chı´nh cu’a ma trˆa.n cu’a da.ng toa`n phu.o.ng d¯ˆe` u du.o.ng, t´u.c la`
Di >0, ∀i = 1, n.
(ii) Da.ng toa`n phu.o.ng ω trˆen R− khˆong gian vector E la` xa´ c d¯i.nh ˆam khi va` chı’ khi tˆ` n ta.i mˆo.t co. so.’ cu’ao E sao trong co. so.’ ˆa´y, (−1)iDi > 0, ∀i = 1, n.
D- i.nh ly´ 4.4 (Luˆa.t qua´n tı´nh). Sˆo´ ca´ c sˆo´ ha.ng co´ hˆe. sˆo´ du.o.ng va` sˆo´ ca´c sˆo´ ha.ng co´ hˆe. sˆo´ ˆam trong da.ng chı´nh t˘a´c cu’a da.ng toa`n phu.o.ng ω la` khˆong thay d¯ˆo’i khi ta thay d¯ˆo’i co. so.’ .
Ba`i tˆa.p.
4.1 Cho f : R×R → R. ´Anh xa. na`o la` a´nh xa. song tuyˆe´n tı´nh: a) f(x, y) = x2+y, b) f(x, y) = x2+y2, c) f(x, y) = 5xy.
4.2 Cho f : R2×R2 →R xa´ c d¯i.nh nhu. sau: ∀x = (x1, x2), y = (y1, y2) ∈ R2: a) f(x, y) = x21+x22+y12+y22, b) f(x, y) = 3x1y1+ 25x2y2
c) f(x, y) = 2x1y2+x2y1, d) f(x, y) = 2x1y1y2
´
Anh xa. na`o la` da.ng song tuyˆe´n tı´nh?
4.3 Cho f : Pn(x)×Pn(x) →R xa´ c d¯i.nh nhu. sau:
∀p(x), q(x) ∈ Pn(x) p(x) = a0+a1x+a2x2+· · ·+anxn q(x) = b0+b1x+b2x2+· · ·+bnxn
f(p(x), q(x)) = a0b0+a1b1+· · ·+anbn.
a) Ch´u.ng minh f la` mˆo.t da.ng song tuyˆe´n tı´nh. b) Tı`m ma trˆa.n cu’a f.
4.4 Cho f : R2×R2 →R xa´ c d¯i.nh nhu. sau: ∀x = (x1, x2), y = (y1, y2) ∈ R2
f(x, y) = 2x1y1+ 3x1y2+ 5x2y2
a) Ch´u.ng minh f la` mˆo.t da.ng song tuyˆe´n tı´nh. b) Tı`m ma trˆa.n cu’a f, rank(f).
4. Da.ng to`an phu
4.5 D- u.a da.ng toa`n phu.o.ng sau d¯ˆay vˆe` da.ng chı´nh t˘a´c, xa´c d¯i.nh phe´p d¯ˆo’i biˆe´n cu˜ theo biˆe´n m´o.i. Tı`m chı’ sˆo´ qua´n tı´nh cu’a da.ng toa`n phu.o.ng. a) x21+ 5x22−4x32 + 2x1x2−4x1x3 b) 4x21+x22+x23−4x1x2+ 4x1x3−3x2x3 c) x1x2+x1x3+x2x3 d) 2x21+ 18x22 + 8x23 −12x1x2+ 8x1x3−27x2x3 e) −12x2 1 −3x2 2−12x2 3+ 12x1x2 −24x1x3+ 8x2x3 f) x1x2 +x2x3+x3x4+x4x1
4.6 Tı`m ca´ c gia´ tri. λ sao cho da.ng toa`n phu.o.ng sau la` xa´c d¯i.nh du.o.ng: a) 5x21+x22+λx23+ 4x1x2−2x1x3−2x2x3
b) 2x21+x22+ 3x32+ 2λx1x2 + 2x1x3
c) x21 +x22+ 5x23+ 2λx1x2−2x1x3+ 4x2x3